   SEARCH HOME Math Central Quandaries & Queries  Question from heather, a student: i have been stuck on this problem for ages The top of a silo is the shape of a hemisphere of diameter 20 ft. if it is coated uniformly with a layer of ice, and if the thickness is decreasing at a rate of 1/4 in/hr, how fast is the volume of ice changing when the ice is 2 inches thick? Hi Heather,

Suppose that the thickness of the ice at time t hours is T(t) inches. The volume of a sphere is 4/3 π r3 so the volume of the hemisphere is 2/3 π (10 × 12)3 = 2/3 π 223 cubic inches. The radius of the hemisphere with the ice layer at time t hours is 10 + T(t) and hence the volume of ice on the hemisphere at time t hours is

V(t) = 2/3 π (10 + T(t))3 - 2/3 π 223 = 2/3 π [(10 + T(t))3 - 223] cubic inches.

Differentiate with respect to t to arrive at an expression for V'(t) involving T(t) and T'(t). Substitute the values of T(t) and T'(t) when T(t) = 2 inches to find V'(t) at this time.

If you need further help with this write back,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.