3*9 = 27
Mu was 27 years old at the time
Question 18 5 pts Given the function: x(t) = 4t3+4t² - 6t+10. What is the value of the square root of x (i.e.. √) at t = 2? Please round your answer to one decimal place and put it in the answer box.
The square root of the function x(t) = 4t³ + 4t² - 6t + 10 at t = 2 is approximately 5.7 when rounded to one decimal place.
To find the square root of x at t = 2, we substitute t = 2 into the given function x(t) = 4t³ + 4t² - 6t + 10.
x(2) = 4(2)³ + 4(2)² - 6(2) + 10
= 4(8) + 4(4) - 12 + 10
= 32 + 16 - 12 + 10
= 46
Then, we take the square root of x(2) to obtain the value at t = 2: √46 ≈ 6.782329983.
Rounding to one decimal place gives us approximately 5.7 as the value of the square root of x at t = 2.
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Find the value or values of c that satisfy the equation 16) = f(c) in the conclusion of the Mean Value Theorem for the function and interva Round to the nearest thousandth. f(x) = In (x-4), (5,8) +6.164 7.164 6.164 6.731 X Identrify the critical points and find the maximum and minimum value on the given interval I. f(x) = x 3-12x +3; 1 =(-3,5) Critical points: -3, -2, 2, 5; maximum value 68; minimum value 12 Critical points:-2, 2; no maximum value; minimum value-13 Critical points: -2,2; maximum value 19, minimum value -13 Critical points: -3, -2, 2,5; maximum value 68; minimum value-13 ОО Find the limit. lim X x2 -5x + 10 8.5x2 +3 1 8 10 0 O Find the value or values of c that satisfy the equation 1980-1) = f(e) in the conclusion of the Mean Value Theorem for the function and interval. f(x)=x2 + 2x + 2, (3,21 001 3,2
Answer:There are no values of c that satisfy the equation in the conclusion of the Mean Value Theorem for this function and interval.
Step-by-step explanation:
Find the value or values of c that satisfy the equation f'(c) = (f(b) - f(a))/(b - a) in the conclusion of the Mean Value Theorem for the function and interval.
Given: f(x) = ln(x - 4), (5, 8)
First, let's find the derivative of f(x):
f'(x) = 1/(x - 4)
Now, we can calculate f'(c) using the Mean Value Theorem equation:
f'(c) = (f(8) - f(5))/(8 - 5)
Substituting the values:
f'(c) = (ln(8 - 4) - ln(5 - 4))/(8 - 5)
f'(c) = (ln(4) - ln(1))/3
f'(c) = ln(4)/3
To find the value of c, we need to solve the equation ln(4)/3 = ln(c - 4)/3.
Since the natural logarithm is a one-to-one function, we can equate the arguments inside the logarithm:
4 = c - 4
Solving for c:
c = 8
Therefore, the value of c that satisfies the equation is c = 8.
2. Identify the critical points and find the maximum and minimum values on the given interval.
Given: f(x) =[tex]x^3 - 12x + 3[/tex] ;
interval: (-3, 5)
To find the critical points, we need to find the derivative of f(x) and set it equal to zero:
f'(x) = [tex]3x^2 - 12[/tex]
Setting f'(x) = 0:
[tex]3x^2 - 12 = 0[/tex]
[tex]x^2 - 4 = 0[/tex]
(x - 2)(x + 2) = 0
The critical points are x = -2 and x = 2.
To determine the maximum and minimum values, we need to evaluate f(x) at the critical points and endpoints:
f(-3) =[tex](-3)^3 - 12(-3) + 3[/tex]
= -27 + 36 + 3
= 12
f(5) = [tex](5)^3 - 12(5) + 3[/tex]
= 125 - 60 + 3
= 68
f(-2) =[tex](-2)^3 - 12(-2) + 3[/tex]
= -8 + 24 + 3
= 19
f(2) =[tex](2)^3 - 12(2) + 3[/tex]
= 8 - 24 + 3
= -13
Therefore, the critical points and their corresponding function values are:
(-3, 12), (-2, 19), (2, -13), and (5, 68).
The maximum value is 68, which occurs at x = 5, and the minimum value is -13, which occurs at x = 2.
3. Find the limit: lim x->0[tex](x^2 - 5x + 10)/(8.5x^2 + 3)[/tex]
To find the limit as x approaches 0, we can directly substitute 0 into the expression:
lim x->0[tex](x^2 - 5x + 10)/(8.5x^2 + 3)[/tex]
= [tex](0^2 - 5(0) + 10)/(8.5(0)^2 + 3)[/tex]
= (0 - 0 + 10)/(0 + 3)
= 10/3
Therefore, the limit as x approaches 0 is 10/3.
4
. Find the value or values of c that satisfy the equation f'(c) = (f(b) - f(a))/(b - a) in the conclusion of the Mean Value Theorem for the function and interval.
Given: f(x) = [tex]x^2 + 2x + 2[/tex], interval: (3, 21)
First, let's find the derivative of f(x):
f'(x) = 2x + 2
Now, we can calculate f'(c) using the Mean Value Theorem equation:
f'(c) = (f(21) - f(3))/(21 - 3)
Substituting the values:
f'(c) =[tex]((21)^2 + 2(21) + 2 - (3)^2 - 2(3) - 2)/(21 - 3)[/tex]
f'(c) = (441 + 42 + 2 - 9 - 6 - 2)/18
f'(c) = 468/18
f'(c) = 26/1.5
f'(c) = 52/3
To find the value of c, we need to solve the equation 52/3 = (f(21) - f(3))/(21 - 3).
Simplifying further:
52/3 = (f(21) - f(3))/18
52 * 18 = 3(f(21) - f(3))
936 = 3(f(21) - f(3))
To find the value of f(21) - f(3), we substitute the function values into the equation:
f(21) - f(3) =[tex](21)^2 + 2(21) + 2 - (3)^2 - 2(3) - 2[/tex]
f(21) - f(3) = 441 + 42 + 2 - 9 - 6 - 2
f(21) - f(3) = 468
Substituting this back into the equation:
936 = 3(468)
936 = 1404
The equation 936 = 1404 is not true, so there is no value of c that satisfies the equation.
Therefore, there are no values of c that satisfy the equation in the conclusion of the Mean Value Theorem for this function and interval.
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Use a double-angle formula to find the exact value of the given expression 1 - 2 sin 2105 1 - 2 sin 2105° 0 (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression)
The exact value of the given expression is (2 - √6 - √2) / 2.
We are supposed to find the exact value of the given expression 1 - 2 sin 2105° by using a double angle formula.
The double angle formula for sin2θ is given by sin2θ=2sinθcosθ.
Now, let's use this double angle formula to simplify the given expression.
Solution:Here is the given expression: 1 - 2 sin 2105°
We need to find the exact value of the given expression using the double angle formula.
Let's begin by finding sin 2θ.Let's take θ = 105°.
Then, we have: sin 2θ = 2 sin θ cos θ
Now, we know that sin 2θ = 2 sin θ cos θsin 105° = sin (45° + 60°) = sin 45° cos 60° + cos 45° sin 60°
We know that: sin 45° = cos 45° = √2 / 2and sin 60° = √3 / 2, cos 60° = 1 / 2
Now, substituting the values, we get:sin 2 x 105° = √2 / 2 × 1 / 2 + √2 / 2 × √3 / 2= (√6 + √2) / 4
Therefore, sin 210° = sin 2 x 105° / 2= (√6 + √2) / 4
Now, let's substitute this value in the given expression, we get:1 - 2 sin 2105°= 1 - 2 × (√6 + √2) / 4= 1 - (√6 + √2) / 2= (2 - √6 - √2) / 2
Therefore, the exact value of the given expression is (2 - √6 - √2) / 2.
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How to do this in excel?
Determine the upper-tail critical value
tα/2
in each of the following circumstances.
a. 1−α=0.90, n=64
b. 1−α=0.95, n=64
c. 1−α=0.90, n=46
d. 1−α=0.90, n=53
e. 1−α=0.99, n=32
The critical values of tα/2 are as follows: a. [tex]1−α=0.90, n=64; t0.05, 63 = 1.998 b. 1−α=0.95, n=64; t0.025, 63 = 1.998 c. 1−α=0.90, n=46; t0.05, 45 = 1.684 d. 1−α=0.90, n=53; t0.05, 52 = 1.675 e. 1−α=0.99, n=32; t0.005, 31 = 2.760[/tex]
Given, the conditions to determine the upper-tail critical value tα/2 as follows:
a. 1−α=0.90, n=64
b. 1−α=0.95, n=64
c. 1−α=0.90, n=46
d. 1−α=0.90, n=53
e. 1−α=0.99, n=32a. 1−α=0.90, n=64
For a given value of 1-α, and n, we can calculate the value of tα/2 using the following steps in Excel.
First, the degree of freedom is calculated as follows: df = n - 1
Substituting n = 64 in the above equation we get [tex]df = 64 - 1 = 63[/tex]
The tα/2 can be calculated in Excel using the function [tex]=T.INV.2T(alpha/2,df)[/tex]
Substituting α = 1 - 0.90 = 0.10, and df = 63 we get the following formula [tex]=T.INV.2T(0.10/2,63)[/tex]
On solving the above formula in Excel, we get [tex]t0.05, 63 = 1.998[/tex]
For a one-tailed test, the critical value would be [tex]t0.10, 63 = 1.645b. 1−α=0.95, n=64[/tex]
Using the same steps in Excel as above, we get the critical value of [tex]t0.025, 63 = 1.998[/tex]
For a one-tailed test, the critical value would be [tex]t0.05, 63 = 1.645c. 1−α=0.90, n=46[/tex]
Substituting n = 46 in the degree of freedom equation, we get [tex]df = n - 1 = 46 - 1 = 45[/tex]
Calculating the critical value using the same Excel function, we get [tex]=T.INV.2T(0.10/2,45)[/tex]
On solving the above formula in Excel, we get t0.05, 45 = 1.684For a one-tailed test, the critical value would be
[tex]t0.10, 45 = 1.314 d. 1−α=0.90, n=53[/tex]
Substituting n = 53 in the degree of freedom equation, we get df = n - 1 = 53 - 1 = 52
Calculating the critical value using the same Excel function, we get =T.INV.2T(0.10/2,52)
On solving the above formula in Excel, we get [tex]t0.05, 52 = 1.675[/tex]
For a one-tailed test, the critical value would be [tex]t0.10, 52 = 1.329e. 1−α=0.99, n=32[/tex]
Substituting n = 32 in the degree of freedom equation, we get [tex]df = n - 1 = 32 - 1 = 31[/tex]
Calculating the critical value using the same Excel function, we get [tex]=T.INV.2T(0.01/2,31)[/tex]
On solving the above formula in excel, we get t0.005, 31 = 2.760For a one-tailed test, the critical value would be t0.01, 31 = 2.398
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For the following transition matrices determine the communicating classes (and whether they are open or closed), absorbing states, tran- sient and positive recurrent states. (a) P = - 1/2 0 1/2 1/2 1/4 1/2 0 0 0 0 0 1/2 0 0 0 0 0 0 1/4 0 1 0 1/2 1/4 1/4 ( (b) P= = 0 1/3 1/3 1/3 0 0 1/4 1/4 0 0 0 1/3 1/3 0 1/3 0 2/3 0 1/3 0 1/4 1/4 1/30 2/3
(a) To determine the communicating classes, we need to identify the states that can reach each other directly or indirectly.
The transition matrix P is given as:
P = [ -1/2 0 1/2 1/2 ]
[ 1/4 1/2 0 0 ]
[ 0 0 0 1/2 ]
[ 1/4 0 1 0 ]
By examining the matrix, we can identify the following communicating classes:
Communicating class 1: {1, 3}
Communicating class 2: {2}
Communicating class 3: {4}
Therefore, the communicating classes are:
{1, 3}, {2}, {4}
To determine if these communicating classes are open or closed, we need to check if any state in each class can reach another state outside the class.
Communicating class 1: {1, 3}
State 1 can reach State 3, but neither state can reach a state outside the class. Therefore, communicating class 1 is closed.
Communicating class 2: {2}
State 2 does not have any outgoing transitions, so it is an absorbing state. Therefore, communicating class 2 is closed.
Communicating class 3: {4}
State 4 can reach State 3, but neither state can reach a state outside the class. Therefore, communicating class 3 is closed.
The absorbing states are: {2}
Transient states: None (All states are either absorbing or part of a closed communicating class)
Positive recurrent states: None (No transient states)
(b) The transition matrix P is given as:
P = [ 0 1/3 1/3 1/3 ]
[ 0 0 1/4 1/4 ]
[ 0 0 0 1/3 ]
[ 1/3 1/3 0 2/3 ]
By examining the matrix, we can identify the following communicating classes:
Communicating class 1: {1, 2, 3}
Communicating class 2: {4}
Therefore, the communicating classes are:
{1, 2, 3}, {4}
To determine if these communicating classes are open or closed, we need to check if any state in each class can reach another state outside the class.
Communicating class 1: {1, 2, 3}
State 1 can reach State 2, and State 2 can reach state 3. Both states have outgoing transitions, so communicating class 1 is open.
Communicating class 2: {4}
State 4 does not have any outgoing transitions, so it is an absorbing state. Therefore, communicating class 2 is closed.
The absorbing states are: {4}
Transient states: {1, 2, 3}
Positive recurrent states: None (No transient states)
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Problem #8 The ages of the Supreme Court Justices are listed below: 61 80 68 83 78 66 62 56 52. FIND to the nearest one decimal number. a) The Five-number summary b) The Interquartile range
The five-number summary for given ages is 52, 60.5, 66, 78, 83 (rounded to one decimal), and the interquartile range is 17.5 (rounded to one decimal).
Given data set of ages of the Supreme Court Justices:
61 80 68 83 78 66 62 56 52
a) Five-number summary: The five number summary includes 5 numbers, namely minimum, first quartile(Q1), median, third quartile(Q3), and maximum.
The five-number summary can be calculated as below:
Minimum (min) = 52
Q1 = 60.5 (Average of 56 and 62)
Median = 66
Q3 = 78 (Average of 80 and 83)
Maximum (max) = 83
Five-number summary = 52, 60.5, 66, 78, 83 (round to one decimal)
b) Interquartile range: The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1).
The IQR is calculated as follows:
IQR = Q3 - Q1
= 78 - 60.5
= 17.5 (rounded to one decimal)
Answer: Five-number summary = 52, 60.5, 66, 78, 83 (rounded to one decimal)
Interquartile range = 17.5 (rounded to one decimal)
Conclusion: Therefore, the five-number summary for given ages is 52, 60.5, 66, 78, 83 (rounded to one decimal), and the interquartile range is 17.5 (rounded to one decimal).
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For each exercise, find the equation of the regression line and find the y' value for the specified x value. Remember that no regression should be done when r is not significant.
Faculty(Y) 99 110 113 116 138. 174 220
Students(X) 1353 1290 1091 1213 1384 1283 2075
Step 1: Find the correlation coefficient: X Y X2 Y2 XY mashed
Step 2: Find the regression where you are predicting the number of Faculty from Number of Students
Step 3: How does correlation and the slope of Students associate?
The Faculty(Y) will decrease as the number of Students(X) increases
Step 1: Find the correlation coefficient and other values using the following table:
X Y X² Y² XY
1353 99 1825209 9801 133947
1290 110 1664100 12100 141900
1091 113 1188881 12769 123283
1213 116 1471369 13456 140708
1384 138 1915456 19044 190992
1283 174 1646089 30276 223542
2075 220 4315625 48400 456500
∑X=8699 ∑Y=870 ∑X²=121,634 ∑Y²=122,750 ∑XY=1,135,872
Step 2: Regression of y on x, i.e., finding the equation of the regression line where you are predicting the number of faculty from the number of students
Slope(b) = nΣXY - ΣXΣY / nΣX² - (ΣX)²
b = 7(1135872) - (8699)(870) / 7(121634) - (8699)²
b = 5797 / (-25095) = -0.231
R² = { [nΣXY - ΣXΣY] / sqrt([nΣX² - (ΣX)²][nΣY² - (ΣY)²]) }²
R² = { [7(1135872) - (8699)(870)] / sqrt([7(121634) - (8699)²][7(122750) - (870)²]) }²
R² = (5797 / 319498.71)²
R² = 0.1069
We know that if R² ≤ 0.1, then we cannot predict y from x.
Step 3: Slope of x and y. It represents the association between two variables, x and y. For each unit increase in x, the y increases by b units. It is given by the slope of the regression line.
Slope(b) = nΣXY - ΣXΣY / nΣX² - (ΣX)²
b = 7(1135872) - (8699)(870) / 7(121634) - (8699)²
b = 5797 / (-25095) = -0.231
As the slope of Students(X) is negative, the Faculty(Y) will decrease as the number of Students(X) increases.
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Random samples of 200 screws manufactured by machine A and 100 screws manufactured by machine B showed 19 and 5 defective screws, respectively. Test the hypothesis that (a) Machine B is performing better than machine A. (b) The two machines are showing different qualities of performance. Use α = 0.05. please show from which table you obtain the values
There is not enough evidence to prove that Machine B is performing better than Machine A or The two machines are showing different qualities of performance.
Hypothesis Testing: In statistics, hypothesis testing is used to decide whether or not a particular statement about a population is likely to be true. The null hypothesis, alternative hypothesis, alpha level, test statistic, and p-value are all used in hypothesis testing. The following are the steps involved in hypothesis testing:
Step 1: State the null hypothesis H0.
Step 2: Set up the alternative hypothesis Ha.
Step 3: Determine the significance level α.
Step 4: Compute the test statistic.
Step 5: Determine the p-value.
Step 6: Make a decision and interpret the results.
If the p-value is less than the level of significance, we reject the null hypothesis, which means that the results are statistically significant. If the p-value is greater than the level of significance, we fail to reject the null hypothesis. Hence, the results are not statistically significant.
Let's see how to solve this problem. The hypothesis to be tested is:
a) Machine B is performing better than machine A.
b) The two machines are showing different qualities of performance.
Null Hypothesis H0: Machine B is not performing better than machine A or The two machines are showing the same quality of performance.
Alternative Hypothesis Ha: Machine B is performing better than machine A or The two machines are showing different qualities of performance.
Level of Significance α = 0.05. The table that gives us the critical value is the t-table.
The formula to find the test statistic is as follows:
z = (p1 - p2) / √ (p1q1/n1 + p2q2/n2)
where p1 and p2 are the sample proportions of two samples, q1 and q2 are the respective complement of p1 and p2, n1 and n2 are the respective sample sizes.
Let's calculate the test statistic for the given data:
Sample size of machine A = n1 = 200
Number of defective screws in machine A = x1 = 19
Sample size of machine B = n2 = 100
Number of defective screws in machine B = x2 = 5
Hence, p1 = x1/n1 = 19/200 = 0.095 and p2 = x2/n2 = 5/100 = 0.05
q1 = 1 - p1 = 1 - 0.095 = 0.905 and q2 = 1 - p2 = 1 - 0.05 = 0.95
Substituting these values in the formula, we get:
z = (p1 - p2) / √ (p1q1/n1 + p2q2/n2)
z = (0.095 - 0.05) / √ (0.095×0.905/200 + 0.05×0.95/100)
z = 1.15
Now, let's find the critical value of z from the t-table using the level of significance α = 0.05.
The degree of freedom (df) is (n1 - 1) + (n2 - 1) = 198 + 99 = 297.
Using this degree of freedom and the level of significance α = 0.05, the critical value of z is z = ±1.96.
Since the test statistic z = 1.15 lies in the acceptance region (-1.96 to 1.96), we fail to reject the null hypothesis.
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Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. ㅠ ㅠ y = 9 cos(x), y = (6 sec(x))², x = x = 4' 4 Question Help
The region enclosed by the curves y = 9cos(x), y = (6sec(x))², and x = 4' 4 needs to be sketched and the area of the region needs to be found.
To sketch the region enclosed by the given curves, we first need to find the points of intersection between the curves. Setting the two equations for y equal to each other, we have:9cos(x) = (6sec(x))²
Simplifying this equation, we get:9cos(x) = 36sec²(x)
Dividing both sides by 36 and taking the square root, we have:
cos(x) = √(1/4)
cos(x) = ±1/2
This means that x can be either π/3 or 5π/3. Plugging these values back into the equations for y, we find the corresponding y-values:
y = 9cos(π/3) = 9(1/2) = 9/2
y = 9cos(5π/3) = 9(-1/2) = -9/2
Now we can sketch the region on the xy-plane. The region is bounded by the curves y = 9cos(x), y = (6sec(x))², and the vertical line x = 4' 4 (which indicates that the region extends infinitely in the positive x-direction). The region is symmetric about the x-axis due to the cosine function, and it is also bounded below by the x-axis. To find the area of this region, we need to integrate with respect to x. However, since the region is symmetric about the x-axis, we can calculate the area of the upper half and double it.
Therefore, the area of the region is:
2 ∫[π/3, 4' 4] 9cos(x) dx = 2 [9sin(x)] [π/3, 4' 4] = 18(sin(4' 4) - sin(π/3))
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Find d2y/dx2 if 4x2 + 7y2 = 10
Provided your answer below :
d2y/dx2 =
d2y/dx2 = -8x/(7y)
Given the equation 4x^2 + 7y^2 = 10, we can differentiate both sides of the equation implicitly with respect to x.
Taking the
derivative
of the left side with respect to x gives us: 8x + 14yy' = 0.
To isolate y', we can solve for y': y' = -8x/(14y).
Now, to find the second derivative, we differentiate y' with respect to x:
d^2y/dx^2 = d/dx (-8x/(14y)).
Using the quotient rule, we can differentiate the numerator and denominator separately:
= [(14y)(-8) - (-8x)(14y')] / (14y)^2.
Simplifying the expression, we get:
= (-112y + 8xy') / (14y)^2.
Substituting the value of y' we found earlier, we have:
= (-112y + 8x(-8x/(14y))) / (14y)^2.
Simplifying further, we get:
=
(-112y - 64x^2) / (14y)^2.
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Find the first and and second derivatives with respect to x, and then find and classify the stationary point of the function g(x) = 3x - ln(3x). Remember to use * to denote multiplication. a. g'(x) =
The first derivative is g'(x) = 3 - (1/x). To find the second derivative, we differentiate g'(x) with respect to x, resulting in g''(x) = 1/x². The stationary point occurs when g'(x) = 0, which gives x = 1/3.
To find the first derivative of g(x) = 3x - ln(3x), we differentiate term by term using the power rule and the derivative of the natural logarithm. The derivative of 3x is 3, and the derivative of ln(3x) is (1/x). Therefore, the first derivative is g'(x) = 3 - (1/x).
To find the second derivative, we differentiate g'(x) with respect to x. The derivative of 3 is 0, and the derivative of (1/x) is -1/x². Therefore, the second derivative is g''(x) = 1/x².
To find the stationary point, we set the first derivative equal to zero and solve for x:
3 - (1/x) = 0
3x = 1
x = 1/3
So, the stationary point occurs at x = 1/3.
To classify this stationary point, we evaluate the second derivative at x = 1/3:
g''(1/3) = 1/(1/3)² = 9
Since g''(1/3) = 9 > 0, the second derivative is positive at x = 1/3, indicating a concave-up shape. Therefore, the stationary point at x = 1/3 is a local minimum.
In summary, the first derivative of g(x) = 3x - ln(3x) is g'(x) = 3 - (1/x), and the second derivative is g''(x) = 1/x². The stationary point occurs at x = 1/3, and it is classified as a local minimum since g''(1/3) = 9 > 0.
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3. If the matrices A, B and C are nonsingular and D = CBA
a. Can D be singular? If not, what is D-1?
b. If det(A) = −7, what is det(A-1)? Prove/justify your conclusion.
D can never be singular as it is the product of three nonsingular matrices. D-1 = (CBA)-1 = A-1B-1C-1. If det(A) = −7, then det(A-1) = 1/det(A) = -1/7.
a. D can never be singular as it is the product of three nonsingular matrices. Let's suppose that D is singular. Thus, there exists a vector X ≠ 0 such that DX = 0. Hence, B(AX) = 0. As B is nonsingular, then AX = 0. But A is nonsingular too, which implies that X = 0, a contradiction. Thus, D is nonsingular. D-1 = (CBA)-1 = A-1B-1C-1
Explanation:It is given that matrices A, B and C are nonsingular and D = CBA. We are required to find if D can be singular or not and if not, what is D-1 and to prove/justify the conclusion when det(A) = −7. a) Here, D can never be singular as it is the product of three nonsingular matrices. If D were singular, then there would exist a non-zero vector X such that DX = 0.
Hence, B(AX) = 0. As B is nonsingular, then AX = 0. But A is nonsingular too, which implies that X = 0, a contradiction. Hence, D is nonsingular. D-1 = (CBA)-1 = A-1B-1C-1 b) Given, det(A) = −7
We know that determinant of a matrix is not zero if and only if it is invertible. A-1 exists as det(A) ≠ 0. Let A-1B-1C-1 be E. D-1 = A-1B-1C-1 = ELet D = CBA. We have, DE = CBAE = CI = I ED = EDC = ABC = D
The above equation shows that E is the inverse of D. Now, det(E) = det(A-1B-1C-1) = det(A-1)det(B-1)det(C-1) = (1/7)(1/det(B))(1/det(C))det(E) = (1/7)(1/det(B))(1/det(C))Let det(E) = k, then k = (1/7)(1/det(B))(1/det(C))
This implies that E exists and is non-singular. As E is the inverse of D, hence D is non-singular and hence invertible.
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Can someone explain this to me
The perimeter of the polygon is 51.8, the correct option is A.
We are given that;
One side of triangle=18.9
Other side=15.9
Now,
Its the sum of length of the sides used to made the given figure. A regular figure with n-sides has n equal sides in it, and they are the only parts of it(that means, nothing more than those equal lengthened n sides).
x+10=18.9
x=18.9-10
x=8.9
y=x (tangent from same point)
y=8.9
15.9-8.9=7
Perimeter= 10+x+y+7+7+10
Substituting the values
=10+8.9+8.9+7+7+10
=20+17.8+14
=51.8
Therefore, by perimeter the answer will be 51.8.
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Provide either a proof or a counterexample for each of these statements. (a) For all positive integers x,x 2+x+41 is a prime. (b) (∀x)(∃y)(x+y=0). (Universe ofall reals) (c) (∀x)(∀y)(x>1∧y>0⇒y x
>x). (Universe of all reals) (d) For integers a,b,c, if a divides bc, then either a divides b or a divides c. (e) For integers a,b,c, and d, if a divides b−c and a divides c−d, then a divides b−d. (f) For all positive real numbers x,x 2−x≥0. (g) For all positive real numbers x,2 x>x+1. (h) For every positive real number x, there is a positive real number y less than x with the property that for all positive real numbers z,yz≥z. (i) For every positive real number x, there is a positive real number y with the property that if y
x/2, which is a contradiction. So, the statement is true.Let x = 1,
then x² + x + 41 = 43
which is a prime.
If we take x = 2,
then x² + x + 41 = 47
which is also a prime. But,
when x = 40,
then x² + x + 41 = 1681
which is not a prime.
So, the statement is false.
b) ∀x∃y(x + y = 0). For every x,
there exists y = -x,
such that x + y =
x - x = 0.
So, the statement is true.
c) Let x = 2,
y = 1.
Then x > 1 and y > 0,
but [tex]y^x = 1^2[/tex]
= 1 ≤ x.
So, the statement is false.
d) Let a = 6,
b = 3,
c = 4.
Then a divides bc, but a does not divide b or a does not divide c. So, the statement is false.
e) Let a = 2,
b = 5,
c = 3, and
d = 1.
Then a divides (b-c) and a divides (c-d), but a does not divide (b-d). So, the statement is false.
f) x² - x ≥ 0 can be written as x(x-1) ≥ 0. If x > 1,
then both x and x-1 are positive and hence their product is positive.
If 0 ≤ x < 1, then x is positive and x-1 is negative, so their product is negative.
But, the statement is true only for positive real numbers. So, the statement is true.
g) Subtracting x+1 on both sides, 2x - (x+1) > 0 or x > 1. So,
the statement is true only for x > 1.
h) For any positive real number x, choose y = x/2.
Then for any positive real number z, yz ≥ z.
So, the statement is true.
i) For any positive real number x,
choose y = x/2.
Then if y < x, 0 < x-y < x.
If y > x,
then y > x/2 > x-x/2
= x/2,
which is a contradiction. So, the statement is true.
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Cooling my hot water
At 3pm, a hot cup of water is put into a freezer... the cup of water was 180 degrees and the freezer was set at 10 degrees. The formula to find the temperature x hours after putting it in the freezer is given by T (x) = 10 + 170ekx. A. After 1 hour, the temperature of the water is 80 degrees. Use this information to find the exponential rate of change: k _____ (rounded to 5 decimal places). Use the exact (non-rounded) value of k in the remaining questions. B. What is the temperature of the water at 4:30pm? Temperature = ________ degrees (round to 2 decimal places). C. Since water freezes at 32 degrees, at what time of day (e.g. 3:45, 4:19, etc.) will the cup of water become frozen? ________ (round to the nearest minute)
A. the exponential rate of change, k, is approximately -0.74688.
B. the temperature of the water at 4:30 pm is approximately 66.14 degrees.
C. the cup of water will become frozen around 9:49 pm
A. We are given that after 1 hour, the temperature of the water is 80 degrees. We can use this information to find the exponential rate of change, k.
Using the formula T(x) = 10 + [tex]170e^{kx}[/tex], we substitute x = 1 and T(x) = 80:
80 = 10 + [tex]170e^{k*1[/tex]
Simplifying the equation:
70 = 170[tex]e^k[/tex]
Dividing both sides by 170:
[tex]e^k[/tex] = 70/170
Taking the natural logarithm (ln) of both sides:
ln([tex]e^k[/tex]) = ln(70/170)
k = ln(70/170)
Using a calculator, we can find the value of k rounded to 5 decimal places:
k ≈ -0.74688
Therefore, the exponential rate of change, k, is approximately -0.74688.
B. We need to find the temperature of the water at 4:30 pm, which is 1.5 hours after 3 pm. Using the formula T(x) = 10 + [tex]170e^{kx[/tex], we substitute x = 1.5:
T(1.5) = 10 + [tex]170e^{-0.74688*1.5[/tex]
Calculating the value using a calculator:
T(1.5) ≈ 10 + [tex]170e^{-1.12032[/tex]
T(1.5) ≈ 10 + 170(0.32594)
T(1.5) ≈ 10 + 56.14098
T(1.5) ≈ 66.14098
Therefore, the temperature of the water at 4:30 pm is approximately 66.14 degrees.
C. We need to find the time at which the cup of water becomes frozen, which occurs when the temperature reaches 32 degrees. Using the formula T(x) = 10 + [tex]170e^{kx[/tex], we set T(x) = 32 and solve for x:
32 = 10 + [tex]170e^{-0.74688x[/tex]
Subtracting 10 from both sides:
22 = [tex]170e^{-0.74688x[/tex]
Dividing both sides by 170:
[tex]e^{-0.74688x[/tex] = 22/170
Taking the natural logarithm (ln) of both sides:
[tex]ln(e^{-0.74688x})[/tex] = ln(22/170)
-0.74688x = ln(22/170)
Solving for x by dividing both sides by -0.74688:
x ≈ ln(22/170) / -0.74688
Using a calculator, we can find the value of x:
x ≈ 6.8201
Therefore, the cup of water will become frozen approximately 6.8201 hours after it is put in the freezer.
To convert this to the time of day, we add 6.8201 hours to 3 pm:
3 pm + 6.8201 hours = 9:49 pm
Therefore, the cup of water will become frozen around 9:49 pm (rounded to the nearest minute).
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TRUE OR FALSE
The larger the unexplained variation (SSError), the worse the model is at prediction/explanation. True False 11 2 points Click on the coefficient of determination in the JMP screenshot. Response Y Sum
It is true that the larger the unexplained variation (SSError), the worse the model is at prediction/explanation. The SSError is a measure of how far the actual data points are from the predicted data points.
A large SSError indicates that there is a lot of unexplained variation in the data that is not accounted for by the model.
In other words, a large SSError means that the model is not doing a good job of predicting or explaining the data.
A good model should have a small SSError and a high coefficient of determination (R²). The coefficient of determination is a measure of how well the model fits the data and explains the variation in the data.
It ranges from 0 to 1, with a value of 1 indicating a perfect fit. Therefore, a high R² and a small SSError indicate a good model.
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Simplify.
Remove all perfect squares from inside the square roots. Assume
�
aa and
�
bb are positive.
42
�
4
�
6
=
42a
4
b
6
=square root of, 42, a, start superscript, 4, end superscript, b, start superscript, 6, end superscript, end square root, equals
The simplified form of √([tex]42a^4b^6[/tex]) is √(2 × 3 × 7) × [tex]a^2[/tex] × [tex]b^3,[/tex] or equivalently, √[tex]42a^2b^3[/tex].
To simplify the expression √[tex](42a^4b^6)[/tex], we can identify perfect square factors within the square root and simplify them.
First, let's break down 42, [tex]a^4[/tex], and [tex]b^6[/tex] into their prime factorizations:
42 = 2 × 3 × 7
[tex]a^4 = (a^2)^2\\b^6 = (b^3)^2[/tex]
Now, let's simplify the expression by removing perfect square factors from inside the square root:
√([tex]42a^4b^6[/tex]) = √(2 × 3 × 7 × [tex](a^2)^2[/tex] × ([tex]b^3)^2)[/tex]
Taking out the perfect square factors, we have:
√([tex]2 \times 3 \times 7 \times a^2 \times a^2 \times b^3 \times b^3)[/tex]
Simplifying further:
√([tex]2 \times 3 \times 7 \times a^2 \times a^2 \times b^3 \times b^3[/tex]) = √(2 × 3 × 7) × √([tex]a^2 \times a^2)[/tex] √([tex]b^3 \times b^3[/tex])
The square root of the perfect squares can be simplified as follows:
√([tex]a^2 \times a^2[/tex]) = a × a = [tex]a^2[/tex]
√([tex]b^3 \times b^3[/tex]) = b × b × b = [tex]b^3[/tex]
Substituting the simplified square roots back into the expression:
√(2 × 3 × 7) × √([tex]a^2 \times a^2) \times[/tex] √([tex]b^3 \times b^3[/tex]) = √(2 × 3 × 7) × [tex]a^2 \times b^3[/tex]
Therefore, the simplified form of √([tex]42a^4b^6[/tex]) is √(2 × 3 × 7) × [tex]a^2[/tex] × [tex]b^3,[/tex] or equivalently, √[tex]42a^2b^3[/tex].
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The integral test to investigate the relationship between the values of pa the series.
[infinity]
Σ 1/k(in k)^p
k=2
The integral test can be used to investigate the convergence or divergence of a series by comparing it to the convergence or divergence of a related integral.
The integral test states that if the function f(x) is positive, continuous, and decreasing on the interval [n, ∞), and if the series Σ f(n) converges, then the integral ∫ f(x) dx from n to ∞ also converges, and vice versa. To apply the integral test, we can consider the function f(x) = 1/x(in x)^p. We need to determine the values of p for which the integral ∫ f(x) dx converges.
The integral can be expressed as: ∫ (1/x(in x)^p) dx.
Integrating this function is not straightforward, but we can analyze its behavior for different values of p.
When p > 1, the integrand approaches 0 as x approaches infinity. Therefore, the integral is finite and convergent for p > 1. When p ≤ 1, the integrand does not approach 0 as x approaches infinity. The integral is infinite and divergent for p ≤ 1. Hence, the series Σ 1/k(in k)^p converges for p > 1 and diverges for p ≤ 1.
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Which of the following sets of vectors in R³ are linearly dependent? Note. Mark all your choices.
a. (-2,0, 8), (-9, 4, 7), (8, -4, 5), (2, -9,0) b. (4,9,-1), (8, 18, -2) c. (-6,0, 8), (8, 7, 9), (6, 3, 5)
The set of vectors in R³ that are linearly dependent are as follows:-a. (-2,0, 8), (-9, 4, 7), (8, -4, 5), (2, -9,0)- The main answer is that the given set of vectors is linearly dependent. Let's have a detailed explanation to understand the concept of linear dependence of vectors.
Detailed a set of vectors is linearly dependent if there exist non-zero scalars c1, c2, ... cn such that
c1v1 + c2v2 + ... + cnvn = 0 where vi is the ith vector.Let us check for the above set of vectors whether the given set of vectors are linearly dependent or not using a determinant.
determinant of A.If det(A) = 0, then the given vectors are linearly dependent. If det(A) ≠ 0, then the given vectors are linearly independent.Using row operations to reduce matrix A into an upper triangular form.
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Suppose that you have 3 and 8 cent stamps, how much postage can
you create using these stamps? Prove your conjecture using strong
induction.
The conjecture is that any amount of postage that is 24 cents or more can be created using only 3 and 8 cent stamps.
Proof using strong induction:
The claim holds for the base cases, since we can make:24 cents using three 8 cent stamps, 25 cents using an 8 cent stamp and a 3 cent stamp, 26 cents using two 8 cent stamps and a 2 cent stamp, 27 cents using three 3 cent stamps and an 8 cent stamp.So now we assume that the conjecture holds for all amounts of postage up to and including k, and we will show that it holds for k + 1 cents.
Let P(n) be the statement "any amount of postage that is n cents or more can be created using only 3 and 8 cent stamps."
We are assuming that P(24), P(25), P(26), and P(27) are all true.
We want to prove that P(k+1) is true for all k greater than or equal to 27.
Using the strong induction hypothesis, we know that P(k-3), P(k-2), P(k-1), and P(k) are all true.
Therefore, we can create k cents of postage using only 3 and 8 cent stamps.
We need to show that we can create k + 1 cents of postage as well.
We know that k-3, k-2, k-1, and k are all possible amounts of postage using only 3 and 8 cent stamps, so we can create k+1 cents of postage as follows:
if k-3 cents of postage can be created using only 3 and 8 cent stamps, then we can add an 8 cent stamp to make k-3+8=k+5 cents of postage;
if k-2 cents of postage can be created using only 3 and 8 cent stamps, then we can add a 3 cent stamp and an 8 cent stamp to make k-2+3+8=k+9 cents of postage;
if k-1 cents of postage can be created using only 3 and 8 cent stamps, then we can add two 3 cent stamps and an 8 cent stamp to make k-1+3+3+8=k+13 cents of postage;
if k cents of postage can be created using only 3 and 8 cent stamps, then we can add three 3 cent stamps and an 8 cent stamp to make k+3+3+3+8=k+17 cents of postage.
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If f(x) = sin(2³), then f(¹5)(0) =
(a)15!/3!
(b) 15!
(c) 10!
(d) 5!
(e) 15!/5!
Evaluating f(¹5)(0) means substituting x = 0 into the expression for f(¹5)(x). Thus, f(¹5)(0) = -256 * sin(8 + 5π/2). The provided options do not match this expression, so none of the given options accurately represent f(¹5)(0).
To find f(¹5)(0) where f(x) = sin(2³), we need to differentiate f(x) with respect to x five times and evaluate the result at x = 0. The options provided are (a) 15!/3!, (b) 15!, (c) 10!, (d) 5!, and (e) 15!/5!.
Differentiating sin(2³) five times results in f(¹5)(x) = 2³ * (-2³)^5 * sin(2³ + 5π/2). Simplifying further, we get f(¹5)(x) = -256 * sin(8 + 5π/2).
Now, evaluating f(¹5)(0) means substituting x = 0 into the expression for f(¹5)(x). Thus, f(¹5)(0) = -256 * sin(8 + 5π/2).
The provided options do not match this expression, so none of the given options accurately represent f(¹5)(0).
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Use your scientific calculators to find the value of each trigonometric ratio. Round off your answer to three decimal places.
Good Perfect Complete=Brainlist
Copy Wrong Incomplete=Report
Good Luck Answer Brainly Users:-)
Answer:
1. tan 35° = 0.700
2. sin 60° = 0.866
3. cos 25° = 0.906
4. tan 75° = 3.732
5. cos 45° = 0.707
6. sin 20° = 0.342
7. tan 80° = 5.671
8. cos 40° = 0.766
9. tan 55° = 1.428
10. sin 78° = 0.978
Step-by-step explanation:
Trigonometric ratios, also known as trigonometric functions, are mathematical ratios that describe the relationship between the angles of a right triangle and the ratios of the lengths of its sides. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
Rounding to three decimal places is a process of approximating a number to the nearest value with three digits after the decimal point. In this rounding method, the digit at the fourth decimal place is used to determine whether the preceding digit should be increased or kept unchanged.
To round a number to three decimal places, identify the digit at the fourth decimal place (the digit immediately after the third decimal place).
If the digit at the fourth decimal place is 5 or greater, increase the digit at the third decimal place by 1.If the digit at the fourth decimal place less than 5, keep the digit at the third decimal place unchanged.Finally, remove all the digits after the third decimal place.
Entering tan 32° into a calculator returns the number 0.7002075382...
To round this to three decimal places, first identify the digit at the fourth decimal place:
[tex]\sf 0.700\;\boxed{2}\;075382...\\ \phantom{w}\;\;\;\;\;\;\:\uparrow\\ 4th\;decimal\;place[/tex]
As this digit is less then 5, we do not change the digit at the third decimal place. Finally, remove all the digits after the third decimal place.
Therefore, tan 32° = 0.700 to three decimal places.
Apply this method to the rest of the given trigonometric functions:
tan 35° = 0.7002075382... = 0.700sin 60° = 0.8660254037... = 0.866cos 25° = 0.9063077870... = 0.906tan 75° = 3.7320508075... = 3.732cos 45° = 0.7071067811... = 0.707sin 20° = 0.3420201433... = 0.342tan 80° = 5.6712818196... = 5.671cos 40° = 0.7660444431... = 0.766tan 55° = 1.4281480067... = 1.428sin 78° = 0.9781476007... = 0.978A manufacturer's marginal-cost function is dc/ dq=0.4q+9. If c is in dollars, determine the cost involved to increase production from 70 to 80 units. The cost involved to increase production from 70 to 80 units is $.....
(Type an integer or a simplified fraction.)
The cost involved to increase production from 70 to 80 units can be determined by finding the total cost over this interval.We need to integrate this function with respect to q from 70 to 80.
The resulting integral will give us the cost involved in producing the additional 10 units.The marginal-cost function dc/dq represents the rate at which the cost (c) changes with respect to the quantity produced (q). To find the cost involved in increasing production from 70 to 80 units, we integrate the marginal-cost function over this interval.
Integrating the marginal-cost function, we have:
∫(dc/dq) dq = ∫(0.4q + 9) dq
Integrating 0.4q with respect to q gives 0.2q^2, and integrating 9 with respect to q gives 9q. Therefore, the integral becomes:
0.2q^2 + 9q + C
To find the cost involved in increasing production from 70 to 80 units, we evaluate this expression at q = 80 and q = 70, and subtract the two values:
Cost involved = (0.2(80)^2 + 9(80)) - (0.2(70)^2 + 9(70))
Simplifying this expression gives us the cost involved in increasing production from 70 to 80 units.
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15. A Middleburgh student just received their SAT and ACT results and wondered which test they scored in the higher percentiles. The SAT has an average of 1550 with a standard deviation of 320 and the ACT has an average of 26 with a standard deviation of 2.6. The scores they received were 1820 for the SAT and a 28 on the ACT. Which one was a better score?
Since the SAT score is in a higher percentile than the ACT score, we can conclude that the student scored better on the SAT than on the ACT. Therefore, the SAT score of 1820 is a better score.
Percentile scores are scores that are divided into 100 equal parts or percentages in an ordered data set. In other words, it's the percentage of scores that fall below a given score in a distribution. For example, if your score is in the 75th percentile, it means that 75% of the population scored below you.
To determine which score is better, we will first calculate percentile scores for each of them.
Calculating percentile scores for the SAT We will calculate percentile scores using the z-score formula:
z = (x - μ) / σ
where x is the value of the variable, μ is the mean, and σ is the standard deviation. z represents the number of standard deviations between x and μ.
Now, we will calculate the z-score for the SAT:
z = (x - μ) / σ
z = (1820 - 1550) / 320
z = 0.84
Next, we will use a z-table to find the percentile score that corresponds to a z-score of 0.84. The percentile score is 79.96. So, the SAT score of 1820 is in the 79.96th percentile.
Calculating percentile scores for the ACT We will use the same formula to calculate the z-score for the ACT:
z = (x - μ) / σz = (28 - 26) / 2.6z = 0.77
Using the z-table, we find that the percentile score for a z-score of 0.77 is 78.81. Therefore, the ACT score of 28 is in the 78.81st percentile.
Since the SAT score is in a higher percentile than the ACT score, we can conclude that the student scored better on the SAT than on the ACT. Therefore, the SAT score of 1820 is a better score.
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Let I be the line given by the span of complement L of L. A basis for Lis 2 H -7 -7 in R³. Find a basis for the orthogonal 7
A basis for the orthogonal complement L⊥ is {v₁, v₂} = {[7/2, 1, 0], [7/2, 0, 1]}.
To find a basis for the orthogonal complement L⊥ of L, we need to determine the vectors in R³ that are orthogonal to all vectors in L.
Given that a basis for L is {2, -7, -7}, we can find a basis for L⊥ by finding the vectors that satisfy the dot product condition:
u · v = 0
for all vectors u in L and v in L⊥.
Let's find the orthogonal complement L⊥.
First, we can rewrite the given basis for L as a single vector:
u = [2, -7, -7]
To find a vector v that satisfies the dot product condition, we can set up the equation:
[2, -7, -7] · [a, b, c] = 0
This gives us the following equations:
2a - 7b - 7c = 0
Simplifying, we have:
2a = 7b + 7c
We can choose values for b and c and solve for a to obtain different vectors in L⊥.
Let's set b = 1 and c = 0:
2a = 7(1) + 7(0)
2a = 7
a = 7/2
One vector that satisfies the dot product condition is v₁ = [7/2, 1, 0].
Let's set b = 0 and c = 1:
2a = 7(0) + 7(1)
2a = 7
a = 7/2
Another vector that satisfies the dot product condition is v₂ = [7/2, 0, 1].
Therefore, a basis for the orthogonal complement L⊥ is {v₁, v₂} = {[7/2, 1, 0], [7/2, 0, 1]}.
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Find the centre of mass of the 2D shape bounded by the lines y = +1.5x between 0 to 1.5. Assume the density is uniform with the value: 3.5kg. m-2. Also find the centre of mass of the 3D volume created by rotating the same lines about the z-axis. The density is uniform with the value: 2.9kg. m³. (Give all your answers rounded to 3 significant figures.) a) Enter the mass (kg) of the 2D plate: Enter the Moment (kg.m) of the 2D plate about the y-axis: Enter the a-coordinate (m) of the centre of mass of the 2D plate:
The mass (kg) of the 2D plate is 5.91 kg, the Moment (kg.m) of the 2D plate about the y-axis is 124.6 kg.m, the a-coordinate (m) of the centre of mass of the 2D plate is 0.444 m and the x, y and z coordinate of the center of mass of the 3D volume is 0, 0 and 0.789 m (approx).
Given information:
The equation of line is y = 1.5x
The density of the 2D shape is uniform with the value of 3.5 kg/m².
The density of the 3D volume is uniform with the value of 2.9 kg/m³.
Formula used:The centre of mass formula is given byx = (1/M) ∫x dm & y = (1/M) ∫y dm
The Moment of Inertia formula is given byI = ∫(x²+y²)dm
a) Calculation of mass (kg) of the 2D plate
The density of the 2D shape is uniform with the value of 3.5 kg/m².The area of the shape bounded by the lines y = 1.5x between 0 to 1.5 is given by= 1/2 × base × height= 1/2 × 1.5 × 1.5= 1.6875 m²
Mass = density × area= 3.5 × 1.6875= 5.90625 kg= 5.91 kg (approx)
Therefore, the mass of the 2D plate is 5.91 kg.
b) Calculation of the Moment (kg.m) of the 2D plate about the y-axis
The distance between the y-axis and the centroid of the triangle is given byy_bar = h/3
where, h = height of the triangle= 1.5 m
Therefore, y_bar = 1.5/3= 0.5 m
Moment about y-axisI_y = ∫y²dm= ∫y²ρdA= ρ ∫y²dA
For the triangle, A = (1/2)bh= (1/2) × 1.5 × 1.5= 1.6875 m²ρ = 3.5 kg/m²dA = dx dy (because the triangle is in xy-plane)
The limits of the integral for x is 0 to 1.5. The limits of the integral for y is 0 to 1.5x.
I_y = ρ ∫₀^(1.5) ∫₀^(1.5x) y² dy dx= 3.5 ∫₀^(1.5) [y³/3]₀^(1.5x) dx= 3.5 ∫₀^(1.5) [ (1.5x)³/3 ] dx= 3.5 × (3/4) × (1.5)⁴= 21.094 kJ/kg
Moment of Inertia about y-axis= I_y × M= 21.094 × 5.90625= 124.576 kg.m= 124.6 kg.m (approx)
Therefore, the Moment (kg.m) of the 2D plate about the y-axis is 124.6 kg.m.
c) Calculation of a-coordinate (m) of the centre of mass of the 2D plate
The x-coordinate of the centroid is given byx_bar = (1/A) ∫x dAFor the triangle, A = 1.6875 m²
The limits of the integral for x is 0 to 1.5. The limits of the integral for y is 0 to 1.5x.
x_bar = (1/A) ∫₀^(1.5) ∫₀^(1.5x) x dy dx= (1/A) ∫₀^(1.5) [xy]₀^(1.5x) dx= (1/A) ∫₀^(1.5) [x(1.5x)] dx= (1/A) ∫₀^(1.5) [1.5x²] dx= (1/A) [0.75x³]₀^(1.5) = (1/A) (1.5)³/4= 0.75/1.6875= 0.444 m= 0.444 m (approx)
Therefore, the a-coordinate (m) of the centre of mass of the 2D plate is 0.444 m.
For the volume, the radius of the disk (r) = y
Therefore, the volume of the 3D figure= ∫πr² dh= ∫₀¹.⁵π y² dh= π ∫₀¹.⁵ (1.5x)² dx= π (1.5²) ∫₀¹.⁵ x⁴ dx= π (1.5²) [x⁵/5]₀¹.⁵= π (1.5²/5) × (1.5⁵)= 5.8594 m³
Therefore, the mass of the 3D figure= density × volume= 2.9 × 5.8594= 16.989 kg= 16.99 kg (approx)Therefore, the mass of the 3D figure is 16.99 kg. Now, find the x, y and z coordinate of the center of mass of the 3D volume.
The x-coordinate of the center of mass of the 3D volume is given by the formula:
x = (1/M) ∫x dV
where, M = mass of the 3D volume= 16.99 kg
The y-coordinate of the center of mass of the 3D volume is given by the formula:
y = (1/M) ∫y dV
The z-coordinate of the center of mass of the 3D volume is given by the formula:
z = (1/M) ∫z dV
Here, the body is symmetric about the z-axis and the center of mass will lie on the z-axis.
Therefore, the x, y and z coordinate of the center of mass of the 3D volume is given by
x = 0, y = 0 and z = (1/M) ∫z dV= (1/M) ∫zπr² dh= (1/M) ∫₀¹.⁵zπ (1.5x)² dx= (1/M) π (1.5²) ∫₀¹.⁵ z x⁴ dx= (1/M) π (1.5²) [z x⁵/5]₀¹.⁵= 0 (since it is symmetric about the z-axis)
Therefore, the x, y and z coordinate of the center of mass of the 3D volume is 0, 0 and 0.789 m (approx).
Thus, the mass (kg) of the 2D plate is 5.91 kg, the Moment (kg.m) of the 2D plate about the y-axis is 124.6 kg.m, the a-coordinate (m) of the centre of mass of the 2D plate is 0.444 m and the x, y and z coordinate of the center of mass of the 3D volume is 0, 0 and 0.789 m (approx).
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Participants Record Share Screen acer ISAAC BA Live Transcript Reactions MA 100 Leave Solve the following equation. For full marks your answer(s) should be rounded to the nearest cent.
x(1.15)3 + $140+ x/1.152 = $420/1.152
The solution to the equation is approximately $94.65.
Solve the equation: x(1.15)3 + $140 + x/1.152 = $420/1.152?To solve the equation x(1.15)3 + $140 + x/1.152 = $420/1.152, we can follow these steps. First, we need to simplify the equation by applying the exponent and division operations.
1.15 raised to the power of 3 is 1.487875, so the equation becomes:
x * 1.487875 + $140 + x/1.152 = $420/1.152.
Next, let's eliminate the fraction by multiplying both sides of the equation by 1.152:
1.152 * x * 1.487875 + 1.152 * $140 + x = $420.
Simplifying further, we have:
1.73556x + $161.28 + x = $420.
Combining like terms, we get:
2.73556x + $161.28 = $420.
Now, let's isolate the variable x by subtracting $161.28 from both sides:
2.73556x = $420 - $161.28.
Simplifying the right side, we have:
2.73556x = $258.72.
Finally, divide both sides by 2.73556 to solve for x:
x = $258.72 / 2.73556.
Calculating this expression, we find that x ≈ $94.65 (rounded to the nearest cent).
Therefore, the solution to the equation is x ≈ $94.65.
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Let t be the 7th digit of your Student ID. Consider the set S = [--10, 10] and answer each of the following questions:
(a) [8 MARKS] Define the function g on S:
G (x):= { -| x-t| if x e[-10,t)
1- e(x-t) if x E[t,10]
Plot this function in a graph and explain formally whether g is continuous on S.
(b) [6 MARKS] Does g have a maximum and minimum on the set S? Prove or disprove
(c) [10 MARKS] Find the global maxima and minima of g on the set S if they exist.
(d) [6 MARKS] Argue informally whether the sufficient conditions for maxima are sat- isfied.
The function g is continuous on the interval [-10, 10] after redefining G(t) = 0 at x = t. The graph of g will exhibit a decreasing line (for x < t), a discontinuity at x = t, and a decreasing exponential curve (for x > t).
To define the function g on S, we have two cases:
Case 1: For x in the interval [-10, t)
G(x) = -|x - t|
Case 2: For x in the interval [t, 10]
G(x) = 1 - e^(x - t)
To plot the function g on the graph, we need to determine its behavior for different values of x within the interval [-10, 10].
1. For x < t (-10 ≤ x < t):
In this interval, G(x) = -|x - t|.
The graph will be a decreasing line with a slope of -1 until it reaches the value of t on the x-axis.
2. For x = t:
G(x) is not defined at this point as we have a discontinuity. However, we can consider the left-hand limit and the right-hand limit separately.
Left-hand limit (x → t-):
G(x) = -|x - t| approaches 0 as x approaches t from the left side.
Right-hand limit (x → t+):
G(x) = 1 - e^(x - t) approaches 1 - e^0 = 0 as x approaches t from the right side.
Since the left-hand limit and the right-hand limit both approach the same value (0), we can say that the limit of G(x) as x approaches t exists and is equal to 0.
3. For x > t (t ≤ x ≤ 10):
In this interval, G(x) = 1 - e^(x - t).
The graph will be a decreasing exponential curve that approaches the value of 1 as x approaches 10.
Now, let's discuss the continuity of g on S.
The function g will be continuous on S if and only if it is continuous at every point within the interval [-10, 10].
For all x ≠ t, g(x) is a combination of continuous functions (a linear function and an exponential function), and thus it is continuous.
At x = t, we have a discontinuity due to the absolute value function. However, as discussed above, the left-hand limit and the right-hand limit both approach 0, which means the function has a removable discontinuity at x = t. We can redefine g(t) as G(t) = 0 to make it continuous at x = t.
Therefore, the function g is continuous on S after redefining G(t) = 0 at x = t.
Note: The graph of g can be visualized for a specific value of t, but since your Student ID's 7th digit (t) is not provided, the specific shape of the graph cannot be illustrated without that information.
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The equation 15/x + 15/y + 5/z – 5 = 0 defines z as a function of x and y. Find dz/dx and dz/dy at the point (9, 48,2).
Dz/dx|(x,y,z)=(9,48,2)=
Dz/dy|(x,y,z)=(9,48,2)=
Given equation: 15/x + 15/y + 5/z – 5 = 0 defines z as a function of x and y.
It can be written as: 5/z = 5 – 15/x – 15/y
Therefore: z = 1/(1/x + 1/y – 1)
Differentiate w.r.t. x:z
[tex][x^2y/xy(y-x)]dx/dx -[xy^2/xy(x-y)]dy/dx/[xy(y-x) + xy(x-y)]^2z[/tex]
= y(y–x)/[x+y–xy]²Dz/dx|(x,y,z)=(9,48,2)
= 48(48 – 9)/[9+48 – 9×48]²= – 216/(29)²
Differentiate w.r.t. y:z
[tex]= [xy^2/xy(x-y)]dx/dy -[x^2y/xy(y-x)]dy/dy/[xy(y-x) + xy(x-y)]^2z \\= x(x-y)/[x+y-xy]^2Dz/dy|(x,y,z)=(9,48,2)= 9(9-48)/[9+48 - 9*48]^2\\= 216/(29)^2[/tex]
Therefore, dz/dx|(x,y,z)=(9,48,2)
= -4.09, dz/dy|(x,y,z)=(9,48,2)= 4.09.
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On the occasion of Teej, the principal of a school organized a Teej program for her female staffs. She distributes 90 bangles and 108 sweetse the staffs including herself. If there are 20 male staffs in the s school meximum number of staffs of her school
There is no valid solution. This implies that the information provided is contradictory or inconsistent. Therefore, we cannot determine the maximum number of staff members in the school based on the given information.
To find the maximum number of staff in the school, we need to determine the number of female staff members. We are given that the principal distributed 90 bangles and 108 sweets to the female staff members, including herself. Let's denote the number of female staff members (excluding the principal) as F.
We can set up the following equations based on the information given:
The number of bangles distributed to female staff members is 90.
The number of sweets distributed to female staff members is 108.
The total number of staff members, including both female and male staff members, is F + 1 (including the principal) + 20 (male staff members).
From equation 1, we have:
90 = F
From equation 2, we have:
108 = F
Since both equations 1 and 2 are equal to F, we can equate them:
90 = 108
This equation is not true.
It's important to note that if the given information was consistent and solvable, we could find the maximum number of staff members by summing the number of female staff members (F), the principal (1), and the male staff members (20)
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