The Skimpy Donut I've come across this problem on several calculus tutorials but can't find any solutions for it.  Can someone please explain how to figure these questions out?
Link to "The Skimpy Donut" problem

For question #1 I found the link below that helped me figure it out:
Volume of a Torus: the Washer Method
ANSWER for question #1:


But I can't find any help on how to solve problems 2, 3, and 4 based on the instructions.  


I will let you solve for the volume and surface area any way you choose. Here I'll just give the results using Pappus's centroid theorems. The solutions are
$$
V=2\pi R A=2\pi^2 Rr^2\\
S=2\pi R C=4\pi^2 Rr
$$
where $R$ is the centroid of the revolving circle, and $A=\pi r^2$ and $C=2\pi r$ are its area and circumference, respectively. Here, $R$ and $r$ correspond to $a$ and $b$ in the problem.
Then we find that we can express
$$S=\frac{2V}{r}$$
If $V$ is fixed, then the maximum surface area coincides with smallest radius. In this problem, the smallest radius is equal to $R$, i.e., the doughnut with no hole has the largest surface area. As the hole increases in size (for a fixed volume) the surface area necessarily decreases.
