Find the length of the barrel and the greatest and least values of the diameter.

Question

The shape of a barrel is made by rotating the segment of the curve $y=40-0.002x^2$ between $x=50$ and $x=-50$, around the x-axis. The units are in centimeters.

1. What is the length of the barrel and what are the greatest and least values of the diameter?
2. What is the volume of the barrel?
3. You can change the volume of the barrel by changing the constants $40$, $0.002$, and $50$. What would be the effects of changing each of these? Could you design a shorter barrel with the same diameter at the end?

I do not have the exact answers and the answers below have not yet been validated, thus it could be great if anyone could help with the accuracy of my answers. Thank you!

1. Length of Barrel =$50+50=100cm$, Greatest value of diameter=$(40)(2)=80cm$, Least values of the diameter=$2[40-0.002(50)^2]$=$2(35)$=$70cm$
2. Volume of Barrel= $\int_{-50}^{50} \pi{(40-0.002x^2)^2}$ = $462000cm^3$ (3s.f)
3. Changing $40$ would shift curve vertically. Changing $0.002$ would make the curve either wider or narrower. Changing $50$ would make the length of the barrel bigger or smaller, and thus it would affect the least value of the diameter. No, a shorter barrel would result in a larger diameter at the end. (Edit: Yes, a shorter barrel can be made with the same diameters at the end, but the shape of the barrel would be different.)

Answer 1

For part (3), suppose $x_1=-x_2$ and $x_1<50$. Then $y(x_1)=40-ax_1^2$. I have left the coefficient on the $x^2$ term as a parameter because this is what determines the shape/bow of the barrel. Using your formula for the diameter at the ends, $d=2[40-ax_1^2]=70 \Longrightarrow 40-ax_1^2=35\Longrightarrow \frac{5}{x_1^2}=a$ Using this, you can find any shorter, or longer barrel for that matter, with the same end diameter and middle diameter. I have made a sort of simulation on desmos modelling this. By moving the $x_1$ slider, you can see how the barrel shape would change if the barrel's length were changed. The intersections of the vertical lines and red and blue curves represent the largest cross-section of the barrel. If you want to add lines $y=\pm 35$, it will show that the end diameter stays constant.