Answers
The total distance ran by the wide receiver is given as follows:
195.6 yd.
How to obtain the distance between two points?Suppose that we have two points with coordinates given as follows:
[tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex]
Then the formula for the distance between these two points is given as follows:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
For the out route, the distance is given as follows:
[tex]D = \sqrt{(60 - 30)^2+(50 - 20)^2} = 42.43[/tex]
To the end zone, the distance is given as follows:
[tex]D = \sqrt{(60 - 80)^2+(50 + 30)^2} = 82.46[/tex]
From the end zone back to the huddle, the distance is given as follows:
[tex]D = \sqrt{(80 - 30)^2+(-30 - 20)^2} = 70.71[/tex]
Hence the total distance is of:
42.43 + 82.46 + 70.71 = 195.6 yd.
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Related Questions
How many ways can you make change for 70 cents using only nickel dime and quarters
Answers
The number of ways to make a change of 70 cents using only nickel, dime and quarters is given as follows:
15 ways.
How to obtain the number of ways to make the change?The value of each coin is given as follows:
Nickel: $5 cents.Dime: 10 cents.Quarters: 25 cents.Hence the ways to make the change are given as follows:
2 quarters and 2 dimes.2 quarters and 4 nickels.2 quarters, 1 dime and 2 nickels.1 quarter, 1 nickel and 4 dimes.1 quarter, 3 nickels and 3 dimes.1 quarter, 5 nickels and 2 dimes.1 quarter, 7 nickels and 1 dime.7 dimes.6 dimes and 2 nickels.5 dimes and 4 nickels.4 dimes and 6 nickels.3 dimes and 8 nickels.2 dimes and 10 nickels.1 dime and 12 nickels.14 nickels.Which combine to 15 ways.
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Find the volume of a cone that has a radius of 8 inches and a slant height of 10 inches. Round your answer to the nearest tenth of a cubic inch.
Answers
Answer:
401.9 in³------------------------------------
Volume of the cone:
V = πr²h/3We have a slant height and radius. They represent a hypotenuse and one of the legs of the right triangle.
Use Pythagorean to find the other leg, the height h of the cone:
[tex]h = \sqrt{10^2-8^2}=\sqrt{100-64} =\sqrt{36}=6\ in[/tex]Find the volume:
V = 3.14*8²*6/3 = 401.92 ≈ 401.9 in³The volume of the cone is calculated as: 671.0 cubic inches.
How to Find the Volume of a Cone?The volume of a cone can be found using the formula:
V = (1/3) * pi * r^2 * h
Where V is the volume, pi is approximately 3.14, r is the radius of the base, and h is the height (or slant height) of the cone.
In this case, the radius of the base is 8 inches and the slant height is 10 inches. So, the volume of the cone can be found by:
V = (1/3) * pi * 8^2 * 10
V = (1/3) * 3.14 * 64 * 10
V = (1/3) * 201.12 * 10
V = 671.04 cubic inches
Rounding the answer to the nearest tenth of a cubic inch, the volume of the cone is approximately 671.0 cubic inches.
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ana lopez borrow $5000 for 75 days at 8% exact interest find how much interest she must pay on the loan and how much will be die at maturity?
Answers
To find the interest Ana must pay on the loan, we can use the formula:
Interest = Principal x Interest Rate x Time (in days) / 365
What's the math rate?
A rate is a unique ratio where the two words are expressed in several units. For instance, the price is 69 for 12 ounces if a 12-ounce can of maize costs 69. This is not a proportion of two comparable units, like shirts. Cents and ounces are the two dissimilar units in this ratio.
In this case:
Interest = $5000 x 8% x 75 / 365 = $200
So, Ana must pay $200 in interest on the loan.
To find the total amount due at maturity, we add the interest to the principal:
Total Due = Principal + Interest = $5000 + $200 = $5200
So, Ana will need to pay $5200 at maturity.
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Customer 1 and customer 2 each buy products X and Z.
Product Y is never sold to customers who buy product Z.
Conclusion:
Product Y is never sold to customers 1 and 2.
If the assumptions are true, is the conclusion
A) Correct
B) Cannot be determined based on the information available
C) Incorrect
Answers
If the assumptions are true, the conclusion is correct.
What assumptions are true?A statement that is included in an argument as an assumption is one whose truth value is (temporarily) accepted as True. The word "let" is frequently used in mathematics to indicate the introduction of an assumption. Let p, for instance (be true). An extreme circumstance is assumed to solve a problem using the Assumption Method (also known as the Supposition Method) in Singapore Math. Comparing this strategy to the Guess and Check method that primary school pupils learn in Primary 3, it is frequently thought of as a quicker substitute.We are informed that
Each of customers 1 and 2 purchases products X and Z.
Customers who purchase product Z are never offered product Y.
The resulting conclusion is that Customers 1 and 2 are never sold Product Y.
According to the information provided, Customer 1 purchased Products X and Z.
Client 2 purchased Products X and Z.
Customers of product Z are unable to purchase product Y.
Therefore, it is obvious that neither customer can buy product Y when they both purchased Z.
Therefore, the conclusion is correct if the presumptions are true.
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The average temperature during the winter in three states is given.
• Maine: 8.4°C below zero
• Minnesota: 10.9°C below zero •Georgia: 8.8°C
Part A Order these temperatures from least to greatest. Select your answers from the drop-down lists. Least to Greatest:
Part B Explain how you determined the order of the temperatures.
Answers
Answer:
10.9 8.4 8.8
Step-by-step explanation:
Both 10.9C and 8.4C are below zero. We can identify both are in negative temperatures. 8.8C was not labeled as below zero, which indicates a positive unit.
Jay Gatsby categorizes wines into one of three clusters. The centroids of these clusters, describing the average characteristics of a wine in each cluster, are listed in the following table. 1 2 8 Characteristic Cluster 2 3 7 Magnesium Flavanoids Proanthocyanins Colorintensity
Answers
Cluster 1 has average magnesium, flavanoids, and proanthocyanins levels, while Cluster 2 has low levels of these compounds, and Cluster 3 has high levels of these compounds. Colorintensity levels vary across clusters.
Cluster 1 has an average level of magnesium, flavanoids, and proanthocyanins. This means that the levels of these compounds are not particularly high or low. Cluster 2 has a lower level of these compounds than Cluster 1, so wines in this cluster would have a milder taste than Cluster 1. Cluster 3 has a higher level of these compounds than Cluster 1, resulting in a more intense flavor. The colorintensity of wines in each cluster can also vary, with darker or lighter wines depending on the cluster. Gatsby’s categorization system is based on these average characteristics, allowing him to determine the flavor and color of wines in each cluster.
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12feet 4ft what’s the pitch
Answers
The area of the football pitch can be found to be 48 ft ²
How to find the area ?The area of the scaled model of the football pitch can be found by using the formula for the area of a rectangle . This is because the scaled model of the pitch takes the form of a rectangle .
The area would therefore be :
= Length x Width
Length = 12 ft
Width = 4 ft
Area of the football pitch model is ;
= 12 x 4
= 48 ft ²
In conclusion, the area of the pitch is 48 ft ² .
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The full question is:
A scaled model of a football pitch measures 12 feet 4 ft. What's the pitch area?
Suppose that f(x) and g(x) are polynomials of degree 4 and 5 respectively. What is the degree of f(x^3) g(X^2)?
Answers
The degree of a polynomial is the highest exponent of its terms. Since f(x) and g(x) are both polynomials of degree 4 and 5 respectively, the degrees of f(x) and g(x) are 4 and 5 respectively.
The degree of f(x^3) g(X^2) is 14.
The degree of a polynomial is the highest exponent of its terms. Since f(x) and g(x) are both polynomials of degree 4 and 5 respectively, the degrees of f(x) and g(x) are 4 and 5 respectively. When we substitute x^3 for x in f(x) and x^2 for x in g(x), the highest exponent of the terms of f(x^3)g(x^2) is 3*4 + 2*5 = 14. Therefore, the degree of f(x^3) g(x^2) is 14.
The degree of f(x^3) g(x^2) is 14, which is calculated by multiplying the degree of f(x) and g(x) with the exponent of their respective terms. The degree of a polynomial is the highest exponent of its terms. Since f(x) and g(x) are both polynomials of degree 4 and 5 respectively, the degrees of f(x) and g(x) are 4 and 5 respectively.
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students want to verify that the centripetal acceleration of an object undergoing uniform circular motion with tangential speed vv and radius rr can be described by the equation ac
Answers
Yes, this is correct. The centripetal acceleration of an object undergoing uniform circular motion with tangential speed v and radius r can be described by the equation a_c = v^2/r.
The centripetal acceleration of an object undergoing uniform circular motion with tangential speed v and radius r can be calculated using the equation a_c = v^2/r.
The centripetal acceleration of an object undergoing uniform circular motion with tangential speed v and radius r can be described by the equation a_c = v^2/r. This equation shows that the acceleration of an object in uniform circular motion is proportional to the square of the velocity and inversely proportional to the radius of the circle. This equation is a result of Newton's second law, which states that the sum of the forces acting on an object is equal to the mass of the object multiplied by its acceleration. The centripetal acceleration is the acceleration of an object towards the center of the circle and is perpendicular to the direction of motion. In order to calculate the centripetal acceleration of an object, the velocity and radius of the circle must be known. The equation a_c = v^2/r can then be used to calculate the centripetal acceleration of the object.
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HELP ASAP! FIND LINEAR EQUATION PARALLEL TO Y = -3/4X + 1 THROUGH POINTS (8,-6)
Answers
The equation of the line parallel to y = (-3/4)x + 1 and passing through (8, 6) is y = (-3/4)x
What is an equation?An equation is an expression showing the relationship between numbers and variables.
The slope intercept form of a straight line is:
y = mx + b
Where m is the slope and b is the y intercept.
Two lines are parallel if they have the same slope.
Given the line y = (-3/4)x + 1, the slope is -3/4. The line parallel to y = (-3/4)x + 1 would have same slope of -3/4.
The parallel line passes through (8, -6), hence:
y - y₁ = m(x - x₁)
y - (-6) = (-3/4)(x - 8)
y + 6 = (-3/4)x + 6
y = (-3/4)x
The equation of the line is y = (-3/4)x
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1 point 8 divided by 0 point 6
Answers
Answer:
3 is the answer
Step-by-step explanation:
1.8/0.6
example: see the image and follow how to do it
Assignment Scoring
Your best submission for each question part is used for your score.
4.
DETAILS AUFEXC4 10.4.025. 28/100 Submissions Used
Find the minimum or maximum value of the quadratic function.
f(x) = x² - 6x + 9
(-2,11)
x Your answer cannot be understood or graded. More Information
State whether the value is a minimum or a maximum.
minimum
O maximum
Submit Answer
Answers
The quadratic function f(x) = x² - 6x + 9 has a minimum at point (3,0).
What is a quadratic function?
A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of second degree.
The given quadratic function is - f(x) = x² - 6x + 9.
Find the differentiation of f(x) -
dy/dx = d/dx (x² - 6x + 9)
d/dx (x²) - d/dx (6x) + d/dx (9)
2x - 6 + 0
2x - 6 = 0
For stationary values, dy/dx = 0
2x - 6 = 0
2x = 6
x = 6/2
x = 3
At x = 3 we get stationary values.
Now, d²y/dx² = 2x - 6
d/dx (2x - 6)
d/dx (2x) - d/dx (6)
2 - 0 = 0
2 = 0
(d²y/dx²) x=3 = 2 (positive)
Therefore, at x = 3 the function is minimum.
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assuming that the curve is banked and the road is frictionless, determine the bank angle with respect to the horizontal.
Answers
The bank angle for a frictionless curved road can be calculated using the equations of motion.
The equation of motion for a particle in a curved road is given by F = mgsin(θ) where F is the centripetal force, m is the mass of the particle, g is the acceleration due to gravity and θ is the bank angle. Thus, the bank angle can be calculated by rearranging the equation to θ = arcsin(F/mg). For example, if the mass of the particle is 5kg, the centripetal force is 30N and the acceleration due to gravity is 9.8ms-2, then the bank angle would be θ = arcsin(30/5x9.8) = 20.7°. Thus, the bank angle with respect to the horizontal is 20.7°.
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a researcher wants to conduct a study to determine whether a newly developed anti-tardy program is successful. two random groups of 100 students each, identified as control and treatment groups, are formed from 200 students who are repeatedly late to school. both groups receive a set of anti-tardy reading materials and a lecture from a teacher and a tardy-reformed student about the negative impacts of being late. in addition, the treatment group also receives materials from the newly developed program. thirty-three of the 100 students in the control and 35 of the 100 students in the treatment group are no longer late to school is recorded 6 months later. use the results to test the hypotheses h0: p1 = p2 and Ha: p1 ? p2 where p1 represents the proportion of students who succeed in the control program and p2 represents the proportion of students who succeed in the newly developed program.
Answers
To test the hypotheses h0: p1 = p2 and Ha: p1 ≠ p2, we can use a two-sample proportion test. This test compares the proportion of success (no longer being late to school) in the control group (p1) to the proportion of success in the treatment group (p2) to determine if there is a statistically significant difference between the two proportions.
Here are the steps to perform the test:
State the hypotheses:
h0: p1 = p2 (the proportion of success in the control group is equal to the proportion of success in the treatment group)
Ha: p1 ≠ p2 (the proportion of success in the control group is not equal to the proportion of success in the treatment group)
Select a significance level (alpha): usually, it is set as 0.05
Calculate the test statistic:
p = (p1n1 + p2n2) / (n1 + n2)
z = (p1 - p2) / sqrt(p*(1-p)*((1/n1)+(1/n2)))
Find the p-value:
Using the z-score, look up the probability in a standard normal table.
Make a decision:
Compare the p-value to the significance level. If the p-value is less than or equal to the significance level, then reject the null hypothesis. If the p-value is greater than the significance level, then fail to reject the null hypothesis.
Conclusion:
Interpret the results in terms of the research question. If the null hypothesis is rejected, then the results suggest that there is a statistically significant difference in the proportion of success between the control group and the treatment group
The question is incomplete, the complete question is:
__"'A researcher wants to conduct a study to determine whether a newly developed anti-tardy program is successful. Two random groups of 100 students each, identified as control and treatment groups, are formed from 200 students who are repeatedly late to school. Both groups receive a set of anti-tardy reading materials and a lecture from a teacher and a tardy-reformed student about the negative impacts of being late. In addition, the treatment group also receives materials from the newly developed program. Thirty-three of the 100 students in the control and 35 of the 100 students in the treatment group are no longer late to school is recorded 6 months later.
Use the results to test the hypotheses H0: p1 = p2 and Ha: p1 ≠ p2 where p1 represents the proportion of students who succeed in the control program and p2 represents the proportion of students who succeed in the newly developed program.
What can you conclude using the significance level α = 0.05?""__
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Find an expression which represents the difference when (2x + 4) is subtracted
from (4x + 2) in simplest terms.
Answers
Answer:
(-2x+2) is the difference
Step-by-step explanation:
Shown above
Repeated addition:
1 3 + 1 3 + 1 3= a b
a =
b =
Answers
Answer:
[tex]\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1[/tex]
a=3
b=3
Third part
Commutative property
Fourth part
Inverse property
Fifth part
Identity property
Step-by-step explanation:
Answer:
1st part is a= 3
b= 9
Second part
x=16
y= -24
Third part
Commutative property
Fourth part
Inverse property
Fifth part
Identity property
Step-by-step explanation:
A man cover 255 km in 6 hours on scooter at a uniform speed find his speed in kilometres per hour
Answers
Answer:
The man's speed is 42.5 km/h.
You can find this by dividing the distance he traveled (255 km) by the time it took him to travel that distance (6 hours).
255 km / 6 hours = 42.5 km/h.
The formula to find speed is:
Speed = Distance / Time
In this case, the distance is 255 km and the time is 6 hours. So using the formula, we can calculate the speed as:
Speed = 255 km / 6 hours = 42.5 km/h
So the man's speed is 42.5 km/h.
Two 3 x 4 rectangles overlap, as shown below. Find the area of the overlapping region (which is shaded).
Rectangles
The Area of a Rectangle
In geometry, a figure is defined as a rectangle if it is a quadrilateral and it has four right angles and two pairs of parallel sided. The area of such a figure is calculated as A=l w, where l represents the length and w represents the width.
Answers
719/64 unit² is the area of the overlapping region (which is shaded)
Rectangles .
What is a rectangle, exactly?
A rectangle is a sort of quadrilateral with parallel sides that are equal to one another and four vertices that are all 90 degrees apart. Because of this, it is also known as an equiangular quadrilateral.
Because the opposite sides of a rectangle are equal and parallel, it can also be referred to as a parallelogram.
the Pythagorean theorem as follows to calculate the value of x .
( 4 - x )² = 3² + x ²
16 - 8x + x² = 9 + x²
16 - 8x = 9
16 - 9 = 8x
7/8 = x
Thus, the area of the shaded region will be equal to the area of rectangle ABDC minus the area of the two triangles on the same rectangle. The area of rectangle ABDC is equal to
A = l * w
= 4 * 3
A = 12
The area of each of the two triangles is equal to
A = 1/2bh
= 1/2 ( 7/8) ( 7/8 )
= 49/128
The area of the two triangles is equal to
A = 2 * 49/128
A = 49/64
Therefore, the area of the shaded region is equal to
12 - 49/64 = 719/64 unit²
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The area of 755.9ft and 610ft
Answers
Answer: 460,874ft^2
Step-by-step explanation:
The area of a rectangle is calculated by multiplying the length by the width. In this case, the length is 755.9ft and the width is 610ft. Therefore, the area of the rectangle is: 755.9ft * 610ft = 460,874ft^2.
I hope this helps :)
so confused
DUE SOON PLEASE HELP IM SO CONFUSED
Answers
Answer:
1) Mean = $75812.50
Median = $74000
Mode = $73100
2) Mean = 24.5
Median = 24.33333...
Modal class = 29 - 33
Step-by-step explanation:
Definitions
The mode is the most frequently occurring data value.The median is the middle value when all data values are placed in order of size.The mean is the sum of all data values divided by the total number of data values.Question 1Given data:
71500, 74900, 69700, 82300, 78500, 73100, 73100, 83400Mean
[tex]\begin{aligned}\textsf{Mean}&=\dfrac{71500+74900+69700+82300+78500+73100+73100+83400}{8}\\ &= \dfrac{606500}{8}\\&=75812.50\end{aligned}[/tex]
Median
Place the data values in order of size (smallest to largest):
69700, 71500, 73100, 73100, 74900, 78500, 82300, 83400As there is an even number of data values, the median is the mean of the middle two values:
[tex]\implies \sf Median=\dfrac{73100+74900}{2}=74000[/tex]
Mode
The most frequently occurring data value is $73100. Therefore:
[tex]\implies \textsf{Mode} = 73100[/tex]
Question 2With grouped data, we can only estimate the mean and median.
Mean
To find an estimate of the mean, assume that every reading in a class takes the value of the class midpoint.
The number of days is the frequency (f).
Add a class midpoint (x) and an fx column to the table:
[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12} \sf Number\;of\;sales&\textsf{Number of days, $f$}&\textsf{Class midpoint, $x$}&fx\\\cline{1-4}\vphantom{\dfrac12}14-18&5&16&80\\\cline{1-4}\vphantom{\dfrac12}19-23&9&21&189\\\cline{1-4}\vphantom{\dfrac12}24-28&6&26&156\\\cline{1-4}\vphantom{\dfrac12}29-33&10&31&310\\\cline{1-4}\vphantom{\dfrac12}\sf Totals&\sum f=30&&\sum fx=735\\\cline{1-4}\end{array}[/tex]
[tex]\textsf{Mean}=\dfrac{\sum fx}{\sum f}=\dfrac{735}{30}=24.5[/tex]
Median
To find an estimate for the median, use linear interpolation.
Add a cumulative frequency column to the table:
[tex]\begin{array}{|c|c|c|}\cline{1-3}\vphantom{\dfrac12} \sf Number\;of\;sales&\textsf{Number of days}&\textsf{Culmulative frequency}\\\cline{1-3}\vphantom{\dfrac12}14-18&5&5\\\cline{1-3}\vphantom{\dfrac12}19-23&9&14\\\cline{1-3}\vphantom{\dfrac12}24-28&6&20\\\cline{1-3}\vphantom{\dfrac12}29-33&10&30\\\cline{1-3}\end{array}[/tex]
Find which class the median is in.
Since n/2 = 30/2 = 15, there are 15 values less than or equal to the median. This means the median must be in the 24-28 class.
Therefore:
[tex]a_1=m-23.5[/tex][tex]b_1=28.5-23.5=5[/tex][tex]a_2=15-14=1[/tex][tex]b_2=20-14=6[/tex]To find the median, m:
[tex]\begin{aligned}\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2} \implies \dfrac{m-23.5}{5}&=\dfrac{1}{6}\\\\ 6(m-23.5)&=5\\\\6m-141&=5\\\\6m&=146\\\\m&=24.3333...\end{aligned}[/tex]
Therefore, the median is 24.3333...
Mode
The modal class is the class with the highest frequency density.
As all the classes are the same width, this is the class with the highest frequency.
Modal class = 29 - 331.
If
z
y
=
x
, which of the following statements is true?
z - x = y
z - y = x
x · y = z
Answers
Answer:b
Step-by-step explanation:
what is the slope through the line of (-8, -11) and (-1, -5)
Answers
The slope of the line through the points (-8, -11) and (-1, -5) is (y2 - y1)/(x2 - x1) = (-5 - (-11))/(-1 - (-8)) = -6/7 = -0.8571428571428571.
: find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Answers
The point that partitions the line in the ratio of 1:3 is (1.333, 3.333).
Given two points P1(x1,y1) and P2(x2,y2), the point that partitions the directed line between these two points in a given ratio r can be calculated using the following formula:
P(x,y) = (x1 + r(x2-x1), y1 + r(y2-y1))
For example, if we have P1(1,2) and P2(4,6) and we want to partition the line in a ratio of 1:3, the point P(x,y) can be calculated as follows:
P(x,y) = (1 + 1/3(4-1), 2 + 1/3(6-2)) = (1.333,3.333)
Therefore, the point that partitions the line in the ratio of 1:3 is (1.333, 3.333).
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given a list of clothing items [a, b, c, ...] and a list of illegal outfits [[a, b], [a, c], ...], return the number of legal outfit combinations. a combination has to have at least 1 outfit and must not contain any illegal combination.
Answers
The objective of backtracking, an algorithmic methodology, is to find every possible solution to a problem by applying the "brute force" method.
What is backtracking used for?
Crosswords, verbal arithmetic, Sudoku, and many other puzzles that need constraint fulfilment may all be solved using backtracking. For parsing, the knapsack problem, and other combinatorial optimization issues, it is frequently the most practical method.
The objective of backtracking, an algorithmic methodology, is to find every possible solution to a problem by applying the "brute force" method. It entails gradually compiling a collection of each answer. A issue would have limits, so any solutions that don't meet them would be eliminated
.
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perpendicular to line y=1/2x-8 passes through (7,-6) point slope form
Answers
The equation of line is y = -2x + 8 and the slope is m = -2
What is an Equation of a line?The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the equation of line be represented as A
Now , the value of A is
The equation of line perpendicular to the line is y = ( 1/2 )x - 8
Now , the slope of the line m₁ = 1/2
For perpendicular lines , the product of the slopes is -1
So , m₁ x m₂ = -1
So , the value of Slope m₂ = -2
And , the line passes through the point P ( 7 , -6 )
Now , the equation of line is y - y₁ = m ( x - x₁ )
Substituting the values in the equation , we get
y - ( -6 ) = ( -2 ) ( x - 7 )
On simplifying the equation , we get
y + 6 = -2x + 14
Subtracting 6 on both sides of the equation , we get
y = -2x + 8
Hence , the equation of line is y = -2x + 8
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SOLVE BY INDIRECT PROOF 1. If n – m is even, then n^2 – m^2 is also an even.2. If x is odd positive integer then x^2 – 1 is divisible by 4. 3. If x is an odd integer, then 8 is a factor of x^2 – 1. 4. Suppose x, y E Z. If x is even, then xy is even.
Answers
All statements can be proven by indirect proof. If the assumptions of the statement are false, the resulting contradiction proves that the initial assumption is wrong and the statement is true.
1. To prove this statement by indirect proof, assume that n – m is an even number but n^2 – m^2 is an odd number. Since n – m is even, it can be written as n – m = 2k, where k is an integer. This can be substituted into n^2 – m^2 = n^2 – (2k+m)^2 to create n^2 – m^2 = n^2 – 4km – m^2. Since both n and m are integers, 4km is an even number. This means that n^2 – m^2 is an even number, contradicting our assumption that it is an odd number. Therefore, our initial assumption is wrong, and n^2 – m^2 must be an even number if n – m is even.
2. To prove this statement by indirect proof, assume that x is an odd positive integer but x^2 – 1 is not divisible by 4. Since x is an odd positive integer, it can be written as x = 2k + 1, where k is an integer. This can be substituted into x^2 – 1 = (2k + 1)^2 – 1 to create x^2 – 1 = 4k^2 + 4k. Since both k and 4 are integers, 4k^2 is an even number. This means that x^2 – 1 is divisible by 4, contradicting our assumption that it is not divisible by 4. Therefore, our initial assumption is wrong, and x^2 – 1 must be divisible by 4 if x is an odd positive integer.
3. To prove this statement by indirect proof, assume that x is an odd integer but 8 is not a factor of x^2 – 1. Since x is an odd integer, it can be written as x = 2k + 1, where k is an integer. This can be substituted into x^2 – 1 = (2k + 1)^2 – 1 to create x^2 – 1 = 4k^2 + 4k. Since 8 is a factor of 4k^2, it is also a factor of x^2 – 1. This means that 8 is a factor of x^2 – 1, contradicting our assumption that it is not a factor of x^2 – 1. Therefore, our initial assumption is wrong, and 8 must be a factor of x^2 – 1 if x is an odd integer.
4. To prove this statement by indirect proof, assume that x is an even integer but xy is an odd number. Since x is an even integer, it can be written as x = 2k, where k is an integer. This can be substituted into xy = (2k)y to create xy = 2ky. Since both k and y are integers, 2ky is an even number. This means that xy is an even number, contradicting our assumption that it is an odd number. Therefore, our initial assumption is wrong, and xy must be an even number if x is an even integer.
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Find the correct representation for the slope of the tangent line to the curve cosxy + 3sin2x + 2y.
a. sin xy (dx/dy +y) + 6 cos x+2
b. -sin xy (dx/dy +y) + 6 sin x cos x+ (2dy/dx)
c. -sin (xy) (x dy/dx) + 6 sin x + 2
d. sin (xy dy/dx +y) + 6cos x+ 2 dy/dx
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The slope of the tangent line to the curve cos(xy) + 3sin^2(x) + 2y at a point (x,y) is -sin(xy) ( x * dy/dx + y) + 6*sin(x)*cos(x) + 2dy/dx.
So the option b is the correct representation for the slope of the tangent line to the curve.
The correct representation for the slope of the tangent line to the curve cos(xy) + 3sin^2(x) + 2y is c. -sin (xy) (x dy/dx) + 6 sin x + 2.
The slope of a tangent line to a curve at a point (x,y) is given by the derivative of the function with respect to x, evaluated at that point.
So to find the slope of the tangent line to the curve cos(xy) + 3sin^2(x) + 2y, we need to find the derivative of the function with respect to x.
The derivative of cos(xy) with respect to x is
-sin(xy) ( x * dy/dx + y)
The derivative of sin^2(x) with respect to x is
2*sin(x)*cos(x)
The derivative of y with respect to x is
dy/dx
So the derivative of the function with respect to x is:
-sin(xy) ( x * dy/dx + y) + 6*sin(x)*cos(x) + 2dy/dx
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Find the number of ways 66 identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.
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There are 331 ways when 66 identical coins can be separated into three nonempty piles
Let the piles have a, b and c coins, with 0<a <b <c. Then, let b=a+k₁, and
c=b+k₂, such that each ki≥1.
The sum is then a+a+k₁+a+k₁+k₂= 66 ⇒ 3a+2k₁+k₂ = 66.
This is simply the number of positive solutions to the equation
3x+2y+z = 66.
If a = 1, then 2k₁ + k₂ = 63⇒ 1≤k₁ ≤31. Each value of k₁ corresponds to a unique value of k₂, so there are 31 solutions in this case.
Similarly, if a = 2, then 2k₁+k₂= 60⇒ 1≤k₁ ≤29, for a total of 29 solutions in this case.
If a = 3, then 2k₁+ k₁= 57 ⇒ 1 ≤ k₁ ≤28, for a total of 28 solutions.
In general, the number of solutions is just all the numbers that aren't a multiple of 3, that are less than or equal to 31.
We then add our cases to get
=1+2+4+....31=1+2+3+...31-3(1+2+3+...10)
=31*(32)/2-3(55)
=31*16-165
=496-165
=331
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Kazuko says the expressions 5x and 6 - x are equivalent expressions, because you
can substitute 1 for x in both expressions and get the same result. Is Kazuko's
reasoning correct? Explain.
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Yes Kazuko's reasoning is correct the expressions 5x and 6 - x are equivalent expressions
How do you determine whether two phrases are equal?When two expressions can be reduced to a single third expression or when one of the expressions can be expressed in the same way as the other, they are said to be equivalent. When values are replaced for the variable and both expressions yield the same result, you may also tell if two expressions are equal.
X-terms and constants should be combined with any other like terms on either side of the equation. Put the terms in the same sequence, with the x-term generally coming before the constants. The two phrases are equal if and only if each of their terms is the same.
5x and 6 - x
5x + 6 - x
2 (2 x + 3)
4 x + 6
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which point represents the center of the circle shown below?
O A Point M
• B. Point K
• C. Point R
• D. Point T
Answers
Point K is the point that marks the centre of the circle depicted below. This is due to the fact that the centre of a circle is the place from which all of its points are equally far. As a result, Point K is the circle's midpoint and consequently its centre.
What is a Circle?To put it simply, a circle is a spherical shape without any edges or line segments. In geometry, it has the shape of a closed curve. The circle's points are set apart from the center by a specified amount.
A circle is a closed, two-dimensional object in which all points in the plane are equally spaced apart from the center. The symmetry line of reflection is formed by each line that traverses the circle. Additionally, it is rotationally symmetric about the center for all angles.
A circle is a figure with a round shape, all of its points lying on the same plane and all of its points being equally spaced from the circle's center.
The center of the circle can be found by finding the midpoint of the diameter, which passes through points K and T.
The midpoint of the diameter can be found by taking the average of the x and y coordinates of points K and T.
Midpoint of AB = (x1+x2/2 , y1+y2/2)
Midpoint of K and T = ((-2 + 4)/2 , (1 + 7)/2)
Midpoint of K and T = (1 , 4)
Therefore, the center of the circle is point K.
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