Math CentralQuandaries & 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,

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