A rectangular sheet of flexible material has a constant area of

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

Problem

180\text{ cm}^2

. At time

t=0

seconds, the length of the sheet is

12\text{ cm}

and is increasing at a rate of

0.8\text{ cm/s}

. The width is simultaneously changing. Assume all edges remain straight and right angles remain

90^{\circ}

as the sheet is stretched. (1) Derive an expression for

\dfrac{dW}{dt}

, the instantaneous rate of change of the width

W

, in terms of the length

L

and

\dfrac{dL}{dt}

. Do not substitute numerical values at this stage. (2) Using your result from part (a), calculate

\dfrac{dW}{dt}

t=0

. (3) Determine the rate of change of the sheet’s perimeter at

t=0

and indicate whether the perimeter is increasing or decreasing. (4) Interpret your result from part (c) in the context of stretching the sheet. Explain how the sign of

\dfrac{dW}{dt}

influences the perimeter’s behavior.

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