Question

find the rate of change of its area when its length is 75in and its width is 35in

Answer 1

The lenght of the rectangle is increasing at the rate of 8in/sec while its width is decreasing at 3 in/sec. Find the rate of change of it's area when it's length is 75 in and width is 35in

Solution :

Let x be the length and y is the width of the rectangle

It is given that increase length of rectnagle is 8

i.e.,

$\frac{dx}{dt}=8$

It is given that decrease width of rectangle is 3in/sec

i.e.,

$\frac{\text{ dy}}{\text{ dt}}=-3$

Area of rectangle is defined as the product of length and breadth

$\text{ Area = xy}$

Differentiate the area with respect to time t;

$\text{ }\frac{dA}{dt}\text{ = x}\frac{dy}{dt}+y\frac{dx}{dt}$

Substitute the value as x = 75 and y = 35 dx/dt = 8 and dy/dt = -3

$\begin{gathered} \text{ }\frac{dA}{dt}\text{ = x}\frac{dy}{dt}+y\frac{dx}{dt} \\ \text{ }\frac{dA}{dt}=75\times(-3)+35\times8 \\ \text{ }\frac{dA}{dt}=-225+280 \\ \text{ }\frac{dA}{dt}=55 \end{gathered}$

Rate of change of area is 55 unit square

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