Optimization Problem (I hate these)?

Question

Find the largest possible volume of a cylinder inscribed in a sphere of radius 8 (cf. profile view above).

http://i144.photobucket.com/albums/r162/patel748/9.png

We need to maximize the following function r and h:
V= pi*r^2*h
Then we solve for r in terms of h obtaining:
r=
We thus obtain the following formula for V in terms of h alone:
V=
h varies over the interval [a,b], where
a =
b =
V has one stationary point at
h =
We conclude that the maximum possible volume of such a cylinder is:
V =

Answer 1

From your figure: r^2 + (h/2)^2 = R^2
Then: V = pi*r^2*h = pi*h*(R^2 - (h/2)^2)
where (from your picture) a = 0 <= h <= 2*R = b.
To find Vmax we solve dV/dh = 0 which gives h = 2*R/sqrt(3).
The (second derivative of V) < 0 at this point which confirms that this is Vmax. Therefore Vmax = 4*pi*R^3/(3sqrt(3)).