The exact length of the polar curve in this problem is given as follows:
320,597 units.
How to calculate the length of a polar curve?The definition of a polar curve is given as follows:
r(θ).
Then the length of the curve over an interval a ≤ θ ≤ b is given by the definite integral presented as follows:
[tex]L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta[/tex]
The curve in this problem is given as follows:
[tex]r = e^{2\theta}[/tex]
The derivative is given as follows:
[tex]\frac{dr}{d\theta} = 2e^{2\theta}[/tex]
Then the arc length is given by the integral presented as follows:
[tex]L = \int_{0}^{2\pi} \sqrt{e^{4\theta} + 4e^{4\theta}} d\theta[/tex]
[tex]L = \int_{0}^{2\pi} \sqrt{5e^{4\theta}} d\theta[/tex]
[tex]L = \sqrt{5}(\int_{0}^{2\pi} e^{2\theta} d\theta)[/tex]
[tex]L = \frac{\sqrt{5}}{2}(e^{2\theta})|_{\theta = 0}^{\theta = 2\pi}[/tex]
Applying the Fundamental Theorem of Calculus to solve the double integral, the length of the arc is givena s follows:
[tex]L = \frac{\sqrt{5}}{2}(e^{4\pi} - 1)[/tex]
L = 320,597 units.
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Which of the following is not a test of congruence of triangle * AAS test SAS test asa test SSA test?
Answer:
SSA
Step-by-step explanation:
If you have 2 triangles that have 2 congruent sides, and a congruent non-included angle, there is 2 possible values for the third side, so SSA does not work
What is the summation of all the roots of the equation [tex]x^{2022}[/tex] = 1
Answer:
x = 1 or x = -1
Step-by-step explanation:
The only number square root and get 1 is 1 and -1. Because 2022 is an even number, so -1 will work in this case, and the final answer will be positive 1!
If you can please give me a Brainliest, thank you and have a good day!
question content area top part 1 a particular type of wood-frame door is priced at $1480 with a sales tax of $370. find the rate of sales tax.
The sales tax rate for a specific type of wood-frame door priced at $1480 is 25%, or $370.
What is percent?Percentages are just fractions with a denominator of 100. We use the percent sign (%) beside a number to indicate that it is a percentage. For instance, if you answered 75 questions correctly out of 100 on a test (75/100), you would have received a 75%. A percentage is a relative value that represents one hundredth of a quantity. One percent (symbolized 1%) is a hundredth part; thus, 100 percent represents the entire amount and 200 percent specifies twice the given quantity. A percentage is a fraction of a whole expressed as a number between 0 and 100.
Here,
Sale tax=$370
Price of door=$1480
Sales tax percent=370/1480*100
=25%
The rate of sales tax for a particular type of wood-frame door is priced at $1480 is 25% that is $370.
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e throw a fair die three times. if we observe the first throw is a 4, what is the probability that there is at least two 4s in the three throws?
The probability that there is at least two 4s in the three throws is 0.05 or 5%
Probability
In statistics, probability deals with the possibility of happening of the events.
Given,
Here we know that the throw a fair die three times. if we observe the first throw is a 4.
And we need to find the probability that there is at least two 4s in the three throws.
According to the given question, we know that the dice is thrown 3 time.
Then the sample space of the given event is 216.
Here we know that the first throw has 4.
And we need to find the probability of getting two 4s in the 3 throws,
So, the possible number of events are.
=> {444, 441, 442, 443, 445, 446, 414, 424, 434, 454, 464}
So, the total possible number is 11.
Then the probability is,
=> 11/216
=> 0.05 or 5%
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can someone pls help me with this one?
Answer:
y = x + 2
Step-by-step explanation:
Use the slope-intercept form:
y=mx+b
M represents the slope of the line
B represents the y-intercept
If the discriminant, b^2-4ac is ___, a quadratic will have two real roots, two points of intersection with the x-axis.
a) negative
b) positive
c) complex
d) zero
If the discriminant, b² - 4ac is positive
How to complete the statement?From the question, we have the following equation of discriminant
The discriminant (d) is calculated as
d = b² - 4ac
The solutions of a quadratic equation are dependent on the following conditions
If d = 0, the quadratic equation has 1 real solutionIf d < 0, the quadratic equation has imaginary solutionsIf d > 0, the quadratic equation has 2 different real solutionsThis means that "d > 0, the quadratic equation has 2 different real solutions" implies that the discriminant is positive
Hence, the complete statement is (b) positive
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here is the graph of y = 5-x for values of x from 0 to 5. on the same grid, draw the graph of y = x+1
The solution for the system of equations would be x = 2 and y = 3.
The general equation of a straight line is -
[y] = [m]x + [c]where -
[m] → is slope of line which tells the unit rate of change of [y] with respect to [x].
[c] → is the y - intercept i.e. the point where the graph cuts the [y] axis.
We have the graph of y = 5 - x.
Refer to the image of the graph attached. It shows the graph of -
y = 5 - xand
y = x + 1The common point of intersection will be the solution for the system :
x = 2 and y = 3.Therefore, the solution is x = 2 and y = 3.
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Marin writes the functions m(x) = startfraction 3 x over x + 7 endfraction and n(x) = startfraction 7 x over 3 minus x endfraction. Which equation must be true for m(x) and n(x) to be inverse functions?.
Equation which represents the m(x) and n(x) are inverse functions given by Option B.
3[7x/(3 - x)] / [ ( 7x / 3 - x ) + 7 ] =7[3x/(x + 7)] / [ 3 - ( 3x / x + 7 ) ] =x.
As given in the question,
Given functions are
m(x) = 3x /(x+7) and n(x) = 7x/ (3 - x)
Let y = m(x)
y = 3x /(x+7)
Now replace x by y and vice versa:
x = 3y / (y + 7) __(1)
Now substitute (1) in n(x)
n(x) = 7x/ (3 - x)
= 7[3y / (y + 7)] / [ 3 - 3y / (y + 7) ] __(3)
= y
Now n(x) = y
y = 7x/ (3 - x)
Replace x by y and vice versa
x = 7y /( 3 - y ) __(2)
Now , substitute (2) in m(x)
m(x) = 3x /(x+7)
= 3[7y /( 3 - y ) ] / [ 7y /( 3 - y ) + 7 ] __(4)
= y
Replace y by x in (3) and (4)
Equation representing m(x) and n(x) as inverse functions:
3[7x/(3 - x)] / [ ( 7x / 3 - x ) + 7 ] =7[3x/(x + 7)] / [ 3 - ( 3x / x + 7 ) ] =x.
Therefore, equation representing m(x) and n(x) as inverse function is given by :
Option B. 3[7x/(3 - x)] / [( 7x / 3 - x ) + 7] =7[3x/(x + 7)] /[3 - ( 3x / x + 7 )] =x.
The complete question is :
Marin write the functions m(x) = 3x /(x+7) and n(x) = 7x/ (3 - x).
Which equation must be true for m(x) and n(x) to be inverse functions?.
Options are shown in attachment.
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If A inversely proportional to the cube of B, A=64 and B=2.
1). Find the law connecting A and B
2). Find the value of the Constant
Answer:
See below
Step-by-step explanation:
1). A = k / (B^3) <==== this becomes A = 512/(B^3) when you use the answer from 2). (below)
2). When A = 64 and B = 2 :
64 = k / (2^3)
64 = k / 8
k = 64*8 = 512
Point C is 3/4 of the way from point A(-4,-2) to point B(8,6). What are thr cordinates of C
Answer:
C = (5, 4)
Step-by-step explanation:
You want the coordinates of point C, located 3/4 of the way from A(-4, -2) to B(8, 6).
Dividing pointThat point is ...
C = A +3/4(B -A)
C = (-4, -2) + (3/4)((8, 6) -(-4, -2))
C = (-4, -2) + (3/4)(8+4, 6+2) = (-4, -2) + (3/4)(12, 8) = (-4, -2) +(9, 6)
C = (-4 +9, -2 +6)
C = (5, 4)
__
Additional comment
Equivalently,
C = (3B +A)/4 = (3(8, 6) +(-4, -2))/4 = (24 -4, 18 -2)/4 = (5, 4)
The method shown above may be more intuitive. The formula shown here is a result of simplifying the one used above.
When computing the weighted average of the coordinates, the nearest point (B) gets the higher weight (3/4).
if sin (2x) = a and 0 < x < pi/2 then sin x is equal to which of the following and why?
a) a
b) a/2
c) 2a
d) a cos x
e) a sex x
explain pleaseee
The solution to the equation are x=a
How to determine the value of sin (2x) = a ?The given equation is sin (2x) = a and 0 < x < pi/2
Sin(2x) = sin x, 0≤x≤2п ........................... (eq. 1)
Using the trigonometrical formula,
sin(2x) = sin xcosx
Let us subtract the formula from equation 1
2sinxcosx = sinx
Therefore, 2sinxcosx-sinx = 0
Let us remove common terms to get
sinx(2cox-1)=0
Now, sinx=0
x= 0, [tex]\pi[/tex], [tex]2\pi[/tex]
Also, 2cos x =1
This means that cos x = 1/2
By implication, x=cos⁻¹(1/2)
That is to say that x= [tex]\frac{\pi }{3} , \frac{5\pi }{3}[/tex]
In conclusion, the value of [tex]\pi[/tex] is a as a single integer
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How do you simplify 2/5 + 5/20?
The simplified form of the expression 2/5 + 5/20 is 13/20
The expression is
2/5 + 5/20
The expression is the mathematical statement that consist of different variables, numbers and mathematical operator. The mathematical operators are addition, subtraction, division and multiplication
The expression is
2/5 + 5/20
Cross multiply the terms in the expression
2/5 + 5/20 = (2×20) + (5×5) / (5 × 20)
Multiply the terms in the numerator and denominator
= (40 + 25) / 100
Add the terms in the numerator
= 65 /100
Simplify the terms by dividing 5 on the both numerator and denominator
= 13/20
Therefore, the simplified form is 13/20
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21. A virus reproduces by tripling every half hour. If a single virus gets into your system and continues at this rate, how much of it will be in your system after a 24-hour period. (Hint: there would be two instances of tripling after the first hour) 24 hours
Please this is urgent need within 20-30 mins
Answer:
Not 100% sure but I believe the answer is 110592.
Step-by-step explanation:
48^3= 110592
[1/2 Points] DETAILS PREVIOUS ANSWERS SCALCET8 11.10.021. Find the Taylor series for f(x) centered at the given value of a- [Assume that has power series expansion Do not show that Rn(x) 0.] f(x) = In(x) , a = f(x) In(9)
The Taylor series of the function is the sum of derivatives of the function up to infinite.
The Taylor series for a function f(x) centered at a is given by:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
To find the Taylor series for f(x) = In(x) centered at a = In(9), we can plug in the values into the formula above:
In(x) = In(9) + In'(9)(x-In(9)) + In''(9)(x-In(9))^2/2! + In'''(9)(x-In(9))^3/3! + ...
= In(9) + (1/9)(x-In(9)) - (1/81)(x-In(9))^2 + (1/729)(x-In(9))^3 + ...
This is the Taylor series for f(x) = In(x) centered at a = In(9).
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(PLEASE HELP)
Consider circle O with diameter GB. Line AD is tangent to circle O at point A and line DB is tangent to circle O at point B. The measure of ∠GBA is 38°. Chord GC bisects ∠G. Determine each measure
The angles GBD, GBD and G measures 90 , 90, 52 degree respectively.
GB is the diameter given.
AD is the tangent given.
The tangent makes 90 degree angle with diameter.
hence the angle GBD will be : 90°
Similarly, GBA will also be 90°
The angle GBA is given as 38°
Hence the angle G will be = 90 - 38 =52
All points in a plane that are at a specific distance from a specific point, the centre, form a circle. In other words, it is the curve that a moving point in a plane draws to keep its distance from a specific point constant. The radius of a circle is the separation between any point on the circle and its centre.
The radius must typically be a positive integer. A degenerate situation is a circle with radius equal to zero (one point). Except when otherwise specified, this article discusses circles in Euclidean geometry, namely the Euclidean plane.
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What is 1/3 of 18? :)
0.4 (2-q) = 0.2 (q+7)
Answers- A. -3 B. -1 C. 3 D. All real numbers
the ratio of 6th and 7th graders are 11 to 9. if there was 108 7th graders how many 6th graders will there be
The number of the students in the sixth grade will be 132 students.
What are ratio and proportion?A ratio is a group of sequentially ordered numbers a and b expressed as a/b, where b is never equal to zero. When two objects are equal, a statement is said to be proportional.
The proportion of sixth and seventh graders is 11 to 9. In the event that there were 108 seventh graders.
The ratio of the number of students in the sixth and seventh will remain constant.
Let 'x' be the number of students in sixth grade. Then the equation is given as,
x / 108 = 11 / 9
Simplify the equation, then we have
x / 108 = 11 / 9
x = 108 × 11 / 9
x = 12 × 11
x = 132
The number of the students in the sixth grade will be 132 students.
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the heights of a maple tree and a cherry tree currently have a ratio of 5:2. if the maple tree grew 20 centimeters, and 20 centimeters was cut off the top of the cherry tree, the ratio of their heights would be 3:1. before this change, how much taller was the maple tree than the cherry tree, in centimeters?
Before the change maple tree was 240 cm longer than cherry tree when the heights of a maple tree and a cherry tree currently have a ratio of 5:2. if the maple tree grew 20 centimeters, and 20 centimeters was cut off the top of the cherry tree, the ratio of their heights would be 3:1.
Define ratio.One quantity can be expressed as a fraction of another using a ratio, which is a number. Only when the two numbers in a ratio have the same unit can they be compared. Ratios are used to compare two objects.
Given,
Current ratio = 5:2
Let y equal the height of the cherry tree, and x represent the height of a maple tree.
x/y = 5/2
New ratio = 3:1
x+20/y-20 = 3/1
Cross multiplying,
x + 20 = 3(y - 20)
x = 3(y - 20) -20
Equating values in x/y = 5/2
3(y-20) - 20/y = 5/2
5y = 2(3y - 60) -20
5y = 6y - 160
y = 160
Now, for value of x,
Equating values in x = 3(y - 20) -20
x = 3(160 - 20) - 20
x = 480 - 60 - 20
x = 400
For the height,
400 - 160
240
Before the change maple tree was 240 cm longer than cherry tree when the heights of a maple tree and a cherry tree currently have a ratio of 5:2. if the maple tree grew 20 centimeters, and 20 centimeters was cut off the top of the cherry tree, the ratio of their heights would be 3:1.
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simplify (1+√3)(1-√3)
Answer:
- 2
Step-by-step explanation:
the factors of a difference of square are
a² - b² = (a + b)(a - b)
(1 + [tex]\sqrt{3}[/tex] )(1 - [tex]\sqrt{3}[/tex] ) ← are the factors of a difference of squares , then
(1 + [tex]\sqrt{3}[/tex] )(1 - [tex]\sqrt{3}[/tex] )
= 1² - ([tex]\sqrt{3}[/tex] )²
= 1 - 3
= - 2
Solve pls Brainliest
Answer:
The ordered pair that satisfies the system of equations is (7, -1)
Step-by-step explanation:
⭐What is a system of equations?
a system of equations is 2 or more equations that intersect at a point (x,y)When we solve for an ordered pair that satisfies the system of equations, we are finding at what point do the equations intersect.
⭐How can we solve a system of equations?
eliminationsubstitutiongraphingFor this problem, I find elimination to be the easiest. I will demonstrate how to use elimination to solve this system of equations.
⭐What is the elimination method?
the elimination method consists of literally eliminating a variable from both equations to solve for another variableWe have the system: [tex]{ x+7y = 0\\\\2x - 8y = 22[/tex]. Let's eliminate the variable "x".
To do so, multiply the first equation by -2, then add both equations.
[tex]-2(x+7y = 0)\\-2x -14y = 0[/tex]
Now add this new equation with the second equation. This eliminates the x variable.
[tex]-2x - 14y = 0 \\+2x -8y = 22\\\\-22y = 22\\y = -1[/tex]
∴ y = -1
Next, we need to solve for the x variable. To do so, substitute the y variable into one of the equations we added and solve for x.
I will substitute y into the equation [tex]2x - 8y = 22[/tex], but you can also substitute it into [tex]-2x-14y = 0[/tex]
[tex]2x - 8y = 22[/tex]
[tex]2x-8(-1) = 22[/tex]
[tex]2x + 8 = 22[/tex]
[tex]2x = 14[/tex]
[tex]x = 7[/tex]
∴ The ordered pair that satisfies the system of equations is (7, -1)
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Timmy and Johnny's parents are very unfair. This year, Timmy received 5 Christmas presents, but his brother received 19. Their parents say that each year, Timmy will receive one more present than he did the last year, and Johnny will receive one less present that he did the last year. How many years will it take for them to receive the same amount of presents
Answer:
7 Years
Step-by-step explanation:
1 Year: 6,18
2 Years: 7,17
3 Years: 8,16
4 Years: 9,15
5 Years: 10,14
6 Years: 11,13
7 Years: 12,12
For 44 points and due in 59 mins help!, Stewart loves playing Genshin Impact so much that he “forgets” to do his schoolwork. It’s really starting to impact his grades and he has 36 overdue assignments. The semester is almost over, and he really needs to improve his grades. He can realistically complete 4 overdue assignments each day.
Draw a graph to represent this situation.
The number of days required to complete his task is 9 days.
What is a graph?
The graph of a relation R is the set of all points (x, y) in a coordinate plane such that x is related to y through the relation R.
Stewart loves playing Genshin Impact so much that he “forgets” to do his schoolwork. It’s really starting to impact his grades and he has 36 overdue assignments. The semester is almost over, and he really needs to improve his grades. He can realistically complete 4 overdue assignments each day.
So he will complete 4 assignments each day and he has to complete the 36 overdue assignments.
The number of days required to do this = 36/4
= 9 days.
Hence, the number of days required to complete his task is 9 days. and this relation can be expressed on the graph as
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What is the vertex form of the equation? y = – x 2 + 12 x – 4 show steps of how you solved please
The vertex form of the equation is in the form of y = a(x-h)^2 + k.
Step 1: Use the formula below to solve for a.
a = -1/4 * (4 - 12) = -1
Step 2: Use the formula below to solve for h.
h = -b/(2a) = 12/(2(-1)) = -6
Step 3: Use the formula below to solve for k.
k = f(h) = -(-6)^2 + 12(-6) - 4 = 36
Therefore, the vertex form of the equation is y = -(x+6)^2 + 36.
In mathematics, what is the vertex?A vertex, or particular point, is a place where two or more lines or edges meet in a mathematical object. Angles, polygons, polyhedra, and graphs are where vertices are most frequently seen. Nodes and vertices in a graph are the same thing.
What is the vertex location?Recall that a parabola's General Form is y = ax2 + bx + c. The x-coordinate of the vertices, which is x = - b/2a, must first be discovered in order to find the vertex from this form. You will use this number to replace x in the parabola equation once you have determined the x-coordinate of the vertex.
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What is the ordered pair that represents the point (−6, −10) after a reflection over the y-axis? (6, 10) (10, 6) (6, −10) (−10, −6)
The ordered pair that represents the point (- 6, - 10) after a reflection over the y-axis will be;
⇒ (6, - 10)
What is Coordinates?
A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.
Given that;
The point is,
⇒ (- 6, - 10)
Now,
Since, The point is,
⇒ (- 6, - 10)
We know that;
The rule for the reflection over the y-axis is;
⇒ (x, y) → (- x, y)
So, The ordered pair that represents the point (- 6, - 10) after a reflection over the y-axis will be;
⇒ (- (- 6), - 10)
⇒ (6, - 10)
Thus, The ordered pair that represents the point (- 6, - 10) after a reflection over the y-axis will be;
⇒ (6, - 10)
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which category, groceries or dining out, has a higher population mean annual credit card charge? dining out what is the point estimate of the difference between the population means? round to the nearest whole number.
groceries has a higher population mean annual credit card charge
What is population mean?
A population is the complete set group of individuals, whether that group comprises a nation or a group of people with a common characteristic. In statistics, a population is the pool of individuals from which a statistical sample is drawn for a study.
The population mean can be calculated by the sum of all values in the given data/population divided by a total number of values in the given data/population.
Given that Bank of America’s Consumer Spending Survey collected data on annual credit card charges in seven different categories of expenditures: transportation, groceries, dining out, household expenses, home furnishings, apparel, and entertainment (US Airways Attache, December 2003).
Sample size n =42
Mean difference d = 850
Std dev s = 1125
Point estimate of the difference is 0
95% confidence level would be
= 0A ± tcritical * stderrror
= A ± 2.021*173.59
= (-350.83 , 350.83)
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Jaheem invests $10,000 in two different money markets. The first earned 12.5% in one year, the second earned 8.25%. If he earned $925 in interest, how much did he invest in each account?
Answer:
Let X be the amount Jaheem invested in the first money market, and Y be the amount he invested in the second money market. We can set up the following system of equations to represent the given information:
X + Y = 10000 (1)
0.125X + 0.0825Y = 925 (2)
We can solve this system of equations using any method of your choice, such as elimination or substitution.
Using elimination, we can multiply equation (1) by 0.125 and equation (2) by 10,000 to eliminate Y:
1.25X + 1250Y = 1250000
125X + 82500Y = 9250000
Subtracting the first equation from the second gives us:
124X + 82350Y = 8037500
Y = (8037500 - 124X)/82350
Substituting this expression for Y into equation (1) and solving for X gives us:
X + (8037500 - 124X)/82350 = 10000
X = $7,500
Therefore, Jaheem invested $7,500 in the first money market and $2,500 in the second money market.
Step-by-step explanation:
SELF EXPLANATORY
Liam deposits $3,500 in a saving account that pays 1.5% interest, compounded quarterly.
a.Find the balance after a year.
B. Compare the interest compounded annually.
Therefore the balance after a year comes out to be $3552.80 and the balance comes out to be after compounded annually is $3552.50
What is compound interest ?Compound interest is the interest on savings that is computed using both the original principal and the interest accrued over time.
It is believed that Italy in the 17th century was where the idea of "interest on interest" or compound interest first emerged. It will accelerate the growth of a sum more quickly than simple interest, which is only applied to the principal sum.
Money multiplies more quickly through compounding, and the higher the
Here,
The Liam deposits $3,500 in a saving account that pays 1.5% interest,
Thus rate=1.5% and principle amount =3500
Therefore it is compounded quartely ,
The balance after year=
C=P[tex]e^{rt}[/tex]
C=3500*[tex]e^{1.5*4}[/tex]
C=3552.80
Therefore the balance after a year comes out to be $3552.80
Therefore the balance comes out to be after compounded annually is
$3552.50
So there is only $0.30 difference between the amounts
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How to factor x^2-2x
Answer:
x(x-2)
Step-by-step explanation:
1) if f(x) = x² - 1 and g(x) = x + 1, then when x doesnt equal -1, f(x) / g(x) =
helpp
Answer:
f(x) / g(x) = x - 1 where x ≠ -1.
Step-by-step explanation:
f(x) / g(x) = x² - 1 / x + 1
= (x + 1)(x - 1) / (x + 1)
= x - 1.
