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A donut shape solid is generated by revolving a circle $(x-a)^2+y^2=b^2$ around the y-axis. $a$ is the distance from the center of the hole of the donut to the center of the circle revolved, and $b$ is the radius of the circle revolved. I'm trying to compute the minimized surface area and the maximized surface area (if the solid has) with the value of $a$ and $b$, while the volume of the solid doesn't change (which is $90\pi^2$). Thanks a lot if someone can help me :)

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Let's call, more conventional, the small and big radius r and R, where R>=r. Then the volume and area are:

V== 2 Pi^2 r^2 R 
A== 2 Pi^2 r R

You specify V= 90 Pi^2. Then A can be written as a function of only one variable, e.g. r:

A = =V/r
R == V/(2 Pi^2 r^2)

Therefore, the largest value of A of infinity is reached for r->0.

The smallest value of A is reached if r == R:

rmin = V^(1/3)/(2^(1/3) \[Pi]^(2/3))

In your case for V= 90 Pi^2

rmin= 3.55689

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