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gayaneshka [121]
3 years ago
14

A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R (> r) from th

e center of the circle. Find the volume of the torus.
Mathematics
1 answer:
jeyben [28]3 years ago
6 0

Consider a circle with radius r centered at some point (R+r,0) on the x-axis. This circle has equation

(x-(R+r))^2+y^2=r^2

Revolve the region bounded by this circle across the y-axis to get a torus. Using the shell method, the volume of the resulting torus is

\displaystyle2\pi\int_R^{R+2r}2xy\,\mathrm dx

where 2y=\sqrt{r^2-(x-(R+r))^2}-(-\sqrt{r^2-(x-(R+r))^2})=2\sqrt{r^2-(x-(R+r))^2}.

So the volume is

\displaystyle4\pi\int_R^{R+2r}x\sqrt{r^2-(x-(R+r))^2}\,\mathrm dx

Substitute

x-(R+r)=r\sin t\implies\mathrm dx=r\cos t\,\mathrm dt

and the integral becomes

\displaystyle4\pi r^2\int_{-\pi/2}^{\pi/2}(R+r+r\sin t)\cos^2t\,\mathrm dt

Notice that \sin t\cos^2t is an odd function, so the integral over \left[-\frac\pi2,\frac\pi2\right] is 0. This leaves us with

\displaystyle4\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}\cos^2t\,\mathrm dt

Write

\cos^2t=\dfrac{1+\cos(2t)}2

so the volume is

\displaystyle2\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}(1+\cos(2t))\,\mathrm dt=\boxed{2\pi^2r^2(R+r)}

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Subtract. 29.82 − 8.001 The difference is
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21.819 is the difference

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The mayor of a town has proposed a plan for the annexation of a new bridge. A political study took a sample of 1000 voters in th
garri49 [273]

Answer:

z=\frac{0.42 -0.39}{\sqrt{\frac{0.39(1-0.39)}{1000}}}=1.945  

p_v =P(Z>1.945)=0.0259  

If we compare the p value obtained with the significance level given \alpha=0.02 we have p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 2% of significance the proportion of residents who favored annexation is not significantly higher than 0.39.  

Step-by-step explanation:

1) Data given and notation  

n=1000 represent the random sample taken

\hat p=0.42 estimated proportion of residents who favored annexation

p_o=0.39 is the value that we want to test

\alpha=0.02 represent the significance level

Confidence=98% or 0.98

z would represent the statistic (variable of interest)

p_v represent the p value (variable of interest)  

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the proportion is higher than 0.39:  

Null hypothesis:p\leq 0.39  

Alternative hypothesis:p > 0.39  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}} (1)  

The One-Sample Proportion Test is used to assess whether a population proportion \hat p is significantly different from a hypothesized value p_o.

3) Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

z=\frac{0.42 -0.39}{\sqrt{\frac{0.39(1-0.39)}{1000}}}=1.945  

4) Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided \alpha=0.02. The next step would be calculate the p value for this test.  

Since is a right tailed test the p value would be:  

p_v =P(Z>1.945)=0.0259  

If we compare the p value obtained with the significance level given \alpha=0.02 we have p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 2% of significance the proportion of residents who favored annexation is not significantly higher than 0.39.  

4 0
3 years ago
Consider that an = 4n-1 represents a sequence. What is the tenth term? A) 361 B) 512 C) 1,024 D) 4,096 E) 262,144
mel-nik [20]

Hello from MrBillDoesMath!

Answer:

Choice E.

Discussion:

Assuming that n >=1, a(n) = 4n -1, then

a(10) = 4(10) - 1 = 39    

but that doesn't match any of the answers...... hmm...

BUT

If you meant that a(n) = 4^(n-1),  then

a(10) = 4^(10-1) = 4^9 =  262144    

which is Choice E.

Thank you,

MrB

4 0
3 years ago
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