Math CentralQuandaries & Queries


Question from Tracy, a student:

A spherical Tootsie Roll Pop you are sucking on is giving up volume at a steady rate of .8 ml/min. How fast will the radius be decreasing when the Tootsie Roll Pop is 20 mm across?

Hi Tracy,

The volume of the Tootsie Roll Pop and its radius are not constant, they are both functions of time t. Let V(t) be the volume in cubic centimeters and r(t) be the radius in centimeters at time t minutes. The relationship between them is

V(t) = 4/3 π r(t)3.

1 milliliter is 1 cubic centimeter so you know that

r′(t) = -0.8 cc/min.

Differentiate V(t) implicitly with respect to t to find an equation for V'(t) in terms of r(t) and r'(t). Substitute r(t) and V(t) when r(t) = 20/2 = 10 mm. Watch out for the units change. You should probably express the radius in cm rather than mm.


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