Chapter 4.6, Problem 82E

To determine

a)

To show that $S(x)=\pi (r+x)\sqrt{1+{(r-x)}^2}$

To determine

b)

To show that $r<\sqrt{2}$, then $S(x)$ is an increasing function.

To determine

c)

Assume that $r>\sqrt{2}$ show that $S$ has two critical point $x_1<x_2$ in $(0,r)$ and that is a local maximum and $S(x_2)$ is a local minimum

To determine

d)

Conclude that the minimum occurs at $x=0$ or $x_2$

To determine

e)

Find the minimum in the cases $r=1.5\text{and}r=2$

To determine

f)

Challenge let $c=\sqrt{(5+3\sqrt{3}})/4\approx 1.597$ prove that the minimum occurs at $x=0(cone)$