These are both exercises in using thechain rulefor function of two variables.

I take it that "a cylinder at the bottom surrounded by a hemisphere" means that the cylinder has a hemisphere on top of it. The volume of a cylinder of radius r and height h is and the volume of a hemisphere of radius r is . The volume of the figure is the sum of those two: . Find dV/dt in terms of dr/dt and dh/dt. .

Use the chain rule:Let U(x,y) be a differentiable and let (r,θ)be the polar coordinates ( that is, x=rcosθ, y=rsinθ)

Compute Ur and Uθand show that lldedivative of Ull^2= (Ur)^2 +(Uθ)^21/r^2

and