Math CentralQuandaries & Queries


Question from Emily, a student:

a cube of ice (i.e.) each side is of the same length) is melting at a rate such that the length of each side is decreasing at a rate of 5cm per hour. how fast is the volume of the cube decreasing (in cubic cm per hour) at the instant the length of each side is 25cm?

Hi Emily,

Suppose that $t$ is time measured in hours. The side length is a function of time so I am going to write the side length as $s(t)$ cm. The side length is decreasing at the rate of 5 cm per hour so $s'(t) = -5$ cm per hour.

The volume of the cube is given by $V(t) = s(t)^3$ cubic centimeters. Differentiate this expression implicitly with respect to $t$ and then solve for $V'(t)$ when $s(t) = 25$ cm.


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