Mathematics

Problem

A rectangular solid has a square base of side length

s(t)

and height

h(t)

. Because of thermal effects, the base is expanding at

0.04\,\text{m}\!/\text{min}

while the height is contracting at

0.015\,\text{m}\!/\text{min}

. When

s=2.0\,\text{m}

and

h=3.0\,\text{m}

, the solid is used to store a chemical whose volume equals that of the solid. (a) Write the volume

V

of the solid as a function of

s

and

h

. (b) Calculate the instantaneous rate of change of the volume at the specified dimensions. State your answer with correct units. (c) If the rates of change remain constant, explain whether the volume will ever begin to increase after this instant. Justify your response.

Intuitive explanation

This is a related rates problem where two dimensions change simultaneously, and you need to find how the total volume changes as a result. The volume of a rectangular solid with square base is $V = s^2 h$ , so you'll use the product rule and chain rule to differentiate with respect to time. The key insight is that $\frac{dV}{dt}$ depends on both how fast the base is growing and how fast the height is shrinking—each contributes separately to the rate of volume change.

Intuitive explanation

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