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

Answer:

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

Step-by-step 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.

$\texttt{Rate of change of height = }\frac{dh}{dt}=3cm/hr\\\\\texttt{Rate of change of width = }\frac{dw}{dt}=-4cm/hr$

Differentiating area with respect to time,

            $\frac{dA}{dt}=h\frac{dw}{dt}+w\frac{dh}{dt}$

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

          $\frac{dA}{dt}=h\frac{dw}{dt}+w\frac{dh}{dt}\\\\\frac{dA}{dt}=5\times (-4)+9\times 3=-20+27=7cm^2/hr$

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