Question

The length x of the rectangle is decreasing at the rate of 5 cm per minute and the width y is increasing at the rate of 4cm per minute. When x=8 cm and y=6 cm, find the rates of change of the area of the rectangle.

Answer 1

Hint: The change in length x and change in width y is given and by using formula of area of triangle, we will differentiate the area of the rectangle and find the rates of change of area.

Complete step-by-step answer:

Let A be the area of the rectangle. We need to find the rate of change of area w.r.t time when x=8 and y=6 cm i.e.., $\dfrac{{dA}}{{dt}}$ at x=8 and y=6
As we know that the area of rectangle=length*breadth.
$ \Rightarrow A = xy$
Differentiate with respect to time
$ \Rightarrow \dfrac{{dA}}{{dx}} = x\dfrac{{dx}}{{dt}} + y\dfrac{{dy}}{{dt}}$
It is given that rate of change of length i.e. $\dfrac{{dx}}{{dt}} = - 5$ (- sign is because it is decreasing) and it is given that rate of change of breadth i.e. $\dfrac{{dy}}{{dt}} = + 4$ (+ sign is because it is increasing).
$ \Rightarrow \dfrac{{dA}}{{dt}} = x\dfrac{{dx}}{{dt}} + y\dfrac{{dy}}{{dt}} = 4x - 5y$
So when x=8 and y=6
$ \Rightarrow \dfrac{{dA}}{{dt}} = 4x - 5y = 4 \times 8 - 5 \times 6 = 32 - 30 = + 2{\text{ c}}{{\text{m}}^2}$.
Hence area is increasing at the rate of 2sq.cm per minute.

Note: Whenever we face such type of problem the key point we have to remember is that change of area is differentiation of area w.r.t time, so differentiate it and substitute the values which is given we will get the required rate of change of area, if it is coming positive then it is increasing otherwise decreasing.