Chapter 12.4, Problem 70E

Spherical caps The volume of the cap of a sphere of radius r and thickness h is $V=\frac{\pi }{3}{h}^2\left(3r-h\right)$, for 0 ≤ h ≤ 2r.

 a.    Compute the partial derivatives V_h and V_r.

b.    For a sphere of any radius, is the rate of change of volume with respect to r greater when h = 0.2r or when h = 0.8r?

c.    For a sphere of any radius, for what value of h is the rate of change of volume with respect to r equal to 1?

d.    For a fixed radius r, for what value of h (0 ≤ h ≤ 2r) is the rate of change of volume with respect to h the greatest?