Question

The height of a rectangle is increasing at a rate of 3 centimeters per hour and the width of the rectangle is decreasing at a rate of 4 centimeters per hour. At a certain instant, the height is 5 centimeters and the width is 9 centimeters. What is the rate of change of the area of the rectangle at that instant (in square centimeters per hour)?

Answer 1

Rate of change of the area of the rectangle at that instant = 7 cm²/hr.

Explanation:

Area of rectangle = Height x Width

A = hw

The height of a rectangle is increasing at a rate of 3 centimeters per hour and the width of the rectangle is decreasing at a rate of 4 centimeters per hour.

`\texttt{Rate of change of height = }(dh)/(dt)=3cm/hr\\\\\texttt{Rate of change of width = }(dw)/(dt)=-4cm/hr`

Differentiating area with respect to time,

`(dA)/(dt)=h(dw)/(dt)+w(dh)/(dt)`

We need to find rate of change of area when the height is 5 centimeters and the width is 9 centimeters.

`(dA)/(dt)=h(dw)/(dt)+w(dh)/(dt)\\\\(dA)/(dt)=5* (-4)+9* 3=-20+27=7cm^2/hr`

Rate of change of the area of the rectangle at that instant = 7 cm²/hr.