A rectangular solid has a square base of side length and height

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

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.

Related theory

Rates of Change
Related Rates
Rate of change of quantities

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