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2 years ago

Could anyone help I just need to know how to do part b

Mathematics

1 answer:

photoshop1234 [79]2 years ago

5 0

Answer:

k = 408, as shown in your problem statement

Step-by-step explanation:

Compute the integral in the usual way, then compare to the given expression and solve for k. We use the "cylindrical shell" method to find the volume.

The integral that gives the volume is ...

$\displaystyle V=\int_0^1{2\pi x(x-x^3)}\,dx+\int_1^2{2\pi x(x^3-x)}\,dx\\\\=\dfrac{2\pi(1^3-0^3)}{3}-\dfrac{2\pi(1^5-0^5)}{5}+\dfrac{2\pi(2^5-1^5)}{5}-\dfrac{2\pi(2^3-1^3)}{3}\\\\=2\pi\left(\dfrac{5\cdot1^3-3\cdot1^5+3(2^5-1^5)-5(2^3-1^3)}{15}\right)=\dfrac{2\pi}{15}(5-3+3(31)-5(7))\\\\=\dfrac{\pi}{15}(120)=\dfrac{\pi}{15}(k-9(8^{5/3}))\qquad\text{compare to given expression}\\\\120=k-288\qquad\text{multiply by $15/\pi$ and simplify}\\\\k=408\qquad\text{add 288}$

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Step-by-step explanation:

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Part a

And we want a linear model on this way y=mx+b, where m represent the slope and b the intercept. In order to find the slope we have this formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

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And the slope would be:

$m=\frac{-91.225}{14.5875}=-6.254$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{32.5}{4}=8.125$

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