Couple of math questions...?

Question

1. Find the dimensions of the right circular cylinder with greatest volume that can be contained in a sphere of radius 1.

2.
What is the largest (by area) rectangle with sides parallel to the coordinate axes that can be contained in the region of the plane bounded by the graphs of y=0, y=x (1-E^(x/x-1)) and x=1?

Answer 1

Volume=pi r^2 * 2h
h=sqrt(1-r^2)

V=2pi r^2 sqrt(1-r^2)
V'=0
Hence
r^2+r-1=0
r=(-1+sqrt(5))/2

Height of the cylinder =2h
=2(sqrt(1-((-1+sqrt(5))/2)^2))