A cylindrical metal can is being manufactured so that its volume increases at a…

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

Problem

A cylindrical metal can is being manufactured so that its volume increases at a constant rate of

60\,\text{cm}^{3}\!/\text{s}

. At the same time, the height of the can is being reduced at

0.20\,\text{cm}\!/\text{s}

to create a squat profile. At the instant when the height is

10\,\text{cm}

and the radius is

3\,\text{cm}

, both dimensions are changing. (a) Derive an expression for

\dfrac{dr}{dt}

in terms of

r

h

\dfrac{dV}{dt}

, and

\dfrac{dh}{dt}

. (b) Using the values provided, compute

\dfrac{dr}{dt}

at that instant. Include units and round your answer to three decimal places. (c) Determine whether the lateral surface area of the cylinder is increasing or decreasing at that moment. Support your conclusion with appropriate calculations.

Related theory

Rates of Change
Related Rates
Rate of change of quantities

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