Volume and area with Lagrange multipliers

Question

I'm having a lot of trouble solving for these problems. I know I have to Use Lagrange multipliers but I don't know how to apply it. Here is the problem:

Answer 1

For a) We have the equation $$3=\pi r\sqrt{r^2+h^2}$$. We will use this equation to eliminate one variable in $$V=\frac{1}{3}\pi r^2h$$. From the first equation above we get by squaring $$\frac{9}{\pi r^2}-r^2=h^2$$ so $$h=\sqrt{\frac{9}{\pi r^2}-r^2}$$ so $$V(r)=\frac{1}{3}\pi r^2\sqrt{\frac{9}{\pi r^2}-r^2}$$. Now you can compute the first derivative and determine the optimal $r$. For b) Since it is $$V=3$$ given, we can eliminate $h$, so we get $$h=\frac{9}{\pi r^2}$$ so we get $$S(r)=\pi r\sqrt{r^2+\frac{81}{r^2r^4}}$$