After converting the provided integral to polar coordinates, the value of integral is evaluated π/2.
What is polar form?When the Cartesian coordinates (x,y) are expressed in the polar coordinates (r, θ), then this form is called the polar form.
The given integral function in the problem is,
[tex]\int_0^{\sqrt{4}} \int\limits^x_{-x} dydx[/tex]
Let suppose, [tex]x=r\cos\theta[/tex] and [tex]y=r\sin\theta[/tex]. Thus,
[tex]\sin\theta=\dfrac{y}{r}\\\cos\theta=\dfrac{x}{r}[/tex]
Limits are y=x. From the trigonometry, the value of theta in the given triangle can be given as,
[tex]\dfrac{\sin\theta}{\cos\theta}=1\\\tan\theta=1\\\theta=\tan^{-1}1\\\theta=45^o\\\theta=\dfrac{\pi}{4}[/tex]
Similarly, for y=-x the value of angle,
[tex]\theta=-\dfrac{\pi}{4}[/tex]
Thus, the limits of theta are from -π/4 to π/4. From the Pythagoras theorem,
[tex]r^2=x^2+y^2\\r^2=(r\cos\theta)^2+(r\sin\theta)^2\\r^2=r^2(1)[/tex]
Thus, the limits of r is from 0 to 1. Convert the given integral in polar form as,
[tex]\int\limits^{\pi/4}_{-\pi/4} \int_0^{1} dt ds\\\int\limits^{\pi/4}_{-\pi/4} [1-0] dt \\\int\limits^{\pi/4}_{-\pi/4} dt \\\dfrac{\pi}{4}-\left(-\dfrac{\pi}{4} \right) \\\dfrac{\pi}{2} \\[/tex]
Hence, after converting the provided integral to polar coordinates, the value of integral is evaluated π/2.
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To convert the given Cartesian integral to polar coordinates, identify the bounds in polar terms, then rewrite the integral accordingly. After setting up the new limits for r and θ, use the relationship between Cartesian and polar coordinates to express the area element, and integrate step-by-step.
Explanation:To convert the integral ∫4√0∫x-xdydx to polar coordinates and evaluate it, we first need to describe the limits of integration and the region of integration in terms of polar coordinates (r, θ). The given integral ranges over a region bounded by the parabola y = √x and the x-axis from x=0 to x=4. Converted to polar coordinates, this region is bounded by the rays θ = 0 and θ = π/2 and the circles r = 0 and r = 4cos(θ).
So the double integral can be rewritten as ∫π/20∫4cos(θ)0 rdrdθ. To evaluate this integral, we integrate r from 0 to 4cos(θ), then integrate θ from 0 to π/2:
∫π/20 (∫4cos(θ)0 r dr) dθ = ∫π/20 [1/2 r^2]|^{4cos(θ)}_0 dθ = ∫π/20 8cos^2(θ) dθ
Using the double angle formula, cos^2(θ) = (1+cos(2θ))/2, the integral becomes:
8 ∫π/20 (1+cos(2θ))/2 dθ = 4 ∫π/20 (1+cos(2θ)) dθ
This can now be integrated directly to get the final result.
The temperature, T , in degrees Fahrenheit, of a cold yam placed in a hot oven is given by T=f(t) , where t is the time in minutes since the yam was put in the oven.
What is the sign of f'(t)? Why?
What are the units of f'(20)? What is the practical meaning of the statement f' (20)=2?
Answer:
a) positive
b)
[tex]^\circ F\text{ per Minute}[/tex]
c) Interpretation of f'(20)=2
Step-by-step explanation:
We are given the following in the question:
[tex]T=f(t)[/tex]
where T is the temperature in degrees Fahrenheit of a cold yam placed in a hot oven and t is the time in minutes since the yam was put in the oven.
a) sign of f'(t)
f'(t) will represent the rate of change in temperature.
f'(t) will represent the change in temperature of yam when 1 minute has passed since it was kept in oven.
Since the temperature will always increase in oven, f'(t) will have a positive sign.
b) units of f'(20)
Since, f'(t) represent the rate of change in temperature. the unit will be
[tex]\dfrac{\text{degrees Fahrenheit}}{\text{Minute}}[/tex]
That is degrees Fahrenheit per minute.
c) f'(20)=2
f'(20) will tell the change in temperature when 20 minutes have passed after the yam has been kept in oven.
Thus, the given statement means that 20 minutes after the yam was kept for in the oven, the temperature of yam was increasing by 2 degree Fahrenheit per minute.
The sign of f'(t) is positive . The units of f'(20) are degrees Fahrenheit per minute. The statement f'(20)=2 means that at 20 minutes, the temperature of the yam is increasing at a rate of 2°F per minute.
The temperature, T, of a cold yam placed in a hot oven, as a function of time t is given by T=f(t), where t is the time in minutes since the yam was put in the oven.
Sign of f'(t)The sign of f'(t) is positive. This is because, as time t increases, the temperature of the cold yam increases due to the hot environment of the oven.
Units of f'(20)The units of f'(20) are degrees Fahrenheit per minute (°F/min). This is because f'(t) represents the rate of change of temperature with respect to time.
Meaning of f'(20)=2The practical meaning of the statement f'(20)=2 is that at t=20 minutes, the temperature of the yam is increasing at a rate of 2 degrees Fahrenheit per minute.
Rewrite the expression ab dc ef such that each variable is replaced by a different non- zero digit and the value of expression is 1. (The answer may not be unique.)
Answer:
ab dc ef is replaced by 12 43 56
Step-by-step explanation:
ab dc ef are replaced by the position they take when writing in an alphabetical order. a is 1, b is 2, c is 3, and so on.
So, ab dc ed is written as
12 43 56
and
56 - 43 - 12 = 1
Shortly after September 11th 2001, a researcher wanted to determine if the proportion of females that favored war with Iraq was significantly different from the proportion of males that favored war with Iraq. In a sample of 73 females, 28 favored war with Iraq. In a sample of 54 males, 29 favored war with Iraq.
a) Let pF represent the proportion of females that favor the war, pM represent the proportion of males that favor the war. What are the proper hypotheses?
A.)H0: pF = pM versus Ha: pF > pM
B.)H0: pF < pM versus Ha: pF = pM
C.)H0: pF = pM versus Ha: pF < pM
D.)H0: pF = pM versus Ha: pF ≠ pM
b) What is the test statistic? Compute the statistic using male statistics subtracted from female statistics. Give your answer to four decimal places.
c) What is the P-value for the test? Give your answer to four decimal places.
d) Using a 0.01 level of significance, what conclusion should be reached?
A.)The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is less than 0.01.
B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
C.)The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is greater than 0.01.
D.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is less than 0.01.
e) What is the lower endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places.
f) What is the upper endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places.
Answer:
a) D.)H0: pF = pM versus Ha: pF ≠ pM
b) [[tex]z=\frac{0.537-0.384}{\sqrt{0.449(1-0.449)(\frac{1}{73}+\frac{1}{54})}}=1.7138[/tex]
c) [tex]p_v =2*P(Z>1.7138)=0.0866[/tex]
d) B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
e) [tex](0.537-0.384) - 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=-0.0755[/tex]
f) [tex](0.537-0.384) + 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=0.3815[/tex]
Step-by-step explanation:
1) Data given and notation
[tex]X_{M}=28[/tex] represent the number of men that favored war with Iraq
[tex]X_{W}=29[/tex] represent the number of women that favored war with Iraq
[tex]n_{M}=73[/tex] sample of male selected
[tex]n_{W}=54[/tex] sample of female selected
[tex]p_{M}=\frac{28}{73}=0.384[/tex] represent the proportion of men that favored war with Iraq
[tex]p_{W}=\frac{29}{54}=0.537[/tex] represent the proportion of women that favored war with Iraq
[tex]\alpha=0.01[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
Part a
We need to conduct a hypothesis in order to checkif the proportion of females that favored war with Iraq was significantly different from the proportion of males that favored war with Iraq , the system of hypothesis would be:
Null hypothesis:[tex]p_{M} = p_{W}[/tex]
Alternative hypothesis:[tex]p_{M} \new p_{W}[/tex]
The best option is:
D.)H0: pF = pM versus Ha: pF ≠ pM
Part b
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{W}-p_{M}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{28+29}{73+54}=0.449[/tex]
Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.537-0.384}{\sqrt{0.449(1-0.449)(\frac{1}{73}+\frac{1}{54})}}=1.7138[/tex]
Part c
We have a significance level provided [tex]\alpha=0.01[/tex], and now we can calculate the p value for this test.
Since is a one two sided test the p value would be:
[tex]p_v =2*P(Z>1.7138)=0.0866[/tex]
Part d
If we compare the p value and the significance level given [tex]\alpha=0.01[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and the best conclusion would be:
B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
Part e
The confidence interval for the difference of two proportions would be given by this formula
[tex](\hat p_W -\hat p_M) \pm z_{\alpha/2} \sqrt{\frac{\hat W_A(1-\hat p_W)}{n_W} +\frac{\hat p_M (1-\hat p_M)}{n_M}}[/tex]
For the 99% confidence interval the value of [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2=0.005[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=2.58[/tex]
And replacing into the confidence interval formula we got:
[tex](0.537-0.384) - 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=-0.0755[/tex]
Part f
[tex](0.537-0.384) + 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=0.3815[/tex]
The proper hypotheses are H0: pF = pM and Ha: pF ≠ pM. The test statistic is -0.087. The p-value is 0.1651.
Explanation:a) The proper hypotheses for this situation are:
H0: pF = pM
Ha: pF ≠ pM
Hence, the correct option is D.
b) The test statistic is computed by subtracting the male statistic from the female statistic:
Test statistic = pF - pM = 28/73 - 29/54 = -0.087
c) The p-value for the test is computed using the test statistic and the appropriate test statistic distribution. Based on the given values, the p-value is found to be 0.1651.
d) Using a 0.01 level of significance, the conclusion should be: The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the p-value is greater than 0.01. Hence, the correct option is D.
e) The lower endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war is -0.2302.
f) The upper endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war is 0.0566.
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Find a formula for the described function.
A rectangle has perimeter 10 m. Express the area A of the rectangle as a function of the length, L, of one of its sides.
Answer:
The area of the rectangle is [tex]A=5L-L^2[/tex].
Step-by-step explanation:
The perimeter of a rectangle is
[tex]P=2(L+W)[/tex]
where, L is length and W is width.
It is given that perimeter of a rectangle is 10m.
[tex]10=2(L+W)[/tex]
Divide both sides by 2.
[tex]5=L+W[/tex]
Subtract L from both sides.
[tex]5-L=W[/tex]
Area of a rectangle is
[tex]A=L\times W[/tex]
Substitute W=(5-L) in the above formula.
[tex]A=L\times (5-L)[/tex]
[tex]A=5L-L^2[/tex]
Therefore, the area of the rectangle is [tex]A=5L-L^2[/tex].
Final answer:
The area A of a rectangle with a fixed perimeter of 10 meters is expressed as a function of the length L by the formula A(L) = L(5 - L), assuming 0 ≤ L ≤ 5 meters.
Explanation:
The question is asking us to express the area A of a rectangle as a function of the length L, one of its sides, given a fixed perimeter of 10 meters. The perimeter of a rectangle is given by the formula P = 2L + 2W, where L is the length and W is the width. For a rectangle with a perimeter of 10 meters, we have:
2L + 2W = 10
W = (10 - 2L) / 2 = 5 - L
The area A of the rectangle is A = L × W = L(5 - L)
This formula A = L(5 - L) gives the area as a function of the length.A: The rectangle's area is a function of its length, expressed as A(L) = L(5 - L), valid for 0 ≤ L ≤ 5, since the minimum possible width is 0 when the length equals 5, and the maximum possible length is 5 when the width equals 0.
Construct a stem-and-leaf plot of the test scores:67, 72, 86, 75, 89, 89, 87, 90, 99, 100.67, 72, 86, 75, 89, 89, 87, 90, 99, 100. How does the stem-and-leaf plot show the distribution of these data?
Answer:
Stem-and-leaf plot of the test scores is shown below.
Step-by-step explanation:
The given data set is
67, 72, 86, 75, 89, 89, 87, 90, 99, 100
Stem-and-leaf: Leaf is the last term and stem is other term. If a number is 32, then 3 is stem and 2 is leaf.
Stem-and-leaf plot of the test scores is
Stem leaf
6 7
7 2,5
8 6,7,9,9
9 0,9
10 0
The length of the rows are similar to the heights of bars in a histogram; longer rows of the data correspond to higher frequency.
Which of the exponential functions below has a y-intercept of 2?
Answer:
the correct answer is B. g(x)=3^x +1
Step-by-step explanation:
I just took the test
Hope this helps
The exponential function with a y-intercept of 2 is in the form y = 2b^x, where 'a' represents the y-intercept of the function. To find this, set x to 0 in the function, resulting in y = a(1), and thus y = a. If the function's y-intercept is 2, the value of 'a' is 2.
Explanation:To determine which exponential function has a y-intercept of 2, you need to recall the standard form of an exponential function, y = abx, where a is the y-intercept of the function. For an exponential function, the y-intercept occurs when x is 0. Thus, when x = 0, the function takes the form y = ab0, and since anything to the power of 0 is 1, the function simplifies to y = a. Therefore, if a function's y-intercept is 2, it means that the value of a must be 2, resulting in the function y = 2bx.
Additionally, understanding the relationship between exponential and logarithmic functions can be helpful. To rewrite a base number b in terms of natural logarithms, you can use the fact that b = eln(b). For example, 2 = eln(2). This is valuable for solving equations involving exponential growth or decay, especially when a y* calculator button is unavailable.
There are 15 numbers on a list, and the smallest number is changed from 12.9 to 1.29.
a. Is it possible to determine by how much the mean changes? If so, by how much does it change?
b. Is it possible to determine the value of the mean after the change? If so, what is the value?
c. Is it possible to determine by how much the median changes? If so, by how much does it changes?
d. Is it possible to determine by how much the standard deviation changes? If so, by how much does it change?
Answer:
Step-by-step explanation:
There are 15 numbers on a list, and the smallest number is changed from 12.9 to 1.29.
a. Is it possible to determine by how much the mean changes? If so, by how much does it change?
Yes. The total would decrease by 11.61. so mean would decrease by 11.61/15 =0.774
b. Is it possible to determine the value of the mean after the change? If so, what is the value?
No. new men would be old mean - 0.774
c. Is it possible to determine by how much the median changes? If so, by how much does it changes?
Though the smallest number is changed the position does not change.
Hence median will not change.
d. Is it possible to determine by how much the standard deviation changes? If so, by how much does it change?
The original variance would be average of square of deviations of the old mean.
Yes, it possible to determine by how much the mean changes. so, 0.7741 change in mean.
No, it is not possible to determine the value of the mean after the change because the value of mean is unknown before the mean
No, it possible to determine by how much the median changes. so, the value of mean will remain same.
No it is impossible to determine how much the standard deviation changes .
Given that,
There are 15 numbers on a list, and the smallest number is changed from 12.9 to 1.29.
We have to find,
All the statements about mean and median are correct or not.
According to the question,
The total would decrease by 11.61.Then,
Mean would decrease by,
[tex]= \dfrac{11.61}{15} \\\\= 0.774[/tex]1
Yes, it possible to determine by how much the mean changes. so, 0.7741 change in mean.
New men would be old mean - 0.7741.
No, it is not possible to determine the value of the mean after the change because the value of mean is unknown before the mean.
Though the smallest number is changed the position does not change.
Yes, It is possible to determine by how much the median changes because we have 15 numbers in a data set and doesn’t effect the median so the median didn’t change.
Hence median will not change.
No, it possible to determine by how much the median changes. so, the value of mean will remain same.
The original variance would be average of square of deviations of the old mean.No it is impossible to determine how much the standard deviation changes .
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True or false? An integer can contain the following characters: digits, the plus sign, the minus sign, and a decimal point.
Answer:false
Step-by-step explanation:
Integer contains only both positive and negative numbers digits but not decimal points.
Answer:False
Step-by-step explanation:The set of integers Z ={....-3,-2,-1,0,1,2,3....}