The length (cm) and width (cm) of a rectangular metal sheet are both increasing with time

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

Problem

The length xx (cm) and width yy (cm) of a rectangular metal sheet are both increasing with time. At a certain instant the sheet measures x=30 x=30\,cm and y=20 y=20\,cm. The manufacturing requirement is that, at every instant, the width must grow with respect to the length according to the slope dydx=0.6\dfrac{dy}{dx}=0.6. At this instant, sensors show dydt=1.8 cm/s\dfrac{dy}{dt}=1.8\,\text{cm/s}. (i) Determine the rate at which the length should be increasing at this instant to meet the requirement. (ii) Using your value from part (i), calculate the instantaneous rate of change of the perimeter P=2(x+y)P=2(x+y) of the sheet. (iii) Hence find the rate at which the area A=xyA=xy of the sheet is changing at this instant.

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