A storage tank used to hold sand is leaking. The sand forms a conical pile whose height is twice the radius of the base. The radius of the pile increases at the rate of 2 inches per minute. Find the rate of change of volume when the radius is 5 inches.

Answer 1

since h = 2r

v = 1/3 pi r^2 (2r) = 2/3 pi r^3

dv/dt = 2 pi r^2 dr/dt = 2*pi*25*2 = 314.16

This is an unrealistic problem. If sand is running out, it's likely that the flow gradually slows down. But this setup has the radius of the pile steadily growing, meaning that the volume increases at an increasing rate. They must be piling more sand into the tank, so it comes out faster and faster to keep the radius steadily growing.

Answer 2

Find the absolute max and min of f(x,y)=4xy^2-x^2y^2-xy^2 on the region D
where D is bounded by x=0, y=0 and y=-x+6