Question

If length $x$ of a rectangle is decreasing at the rate of 3 cm/minute and the width $y$ is increasing at the rate of 2 cm/minute, when $x=10$ cm and $y=6cm$, find the rates of change of (i) the perimeter, (ii) the area of the rectangle.

Answer 1

Rate of decrease in length is $\frac{dx}{dt}=-3cm/min$

Negative sign shows it is decreasing.

Rate of increase in width is $\frac{dy}{dt}=2cm/min$

Perimeter of the rectangle is $P=2x+2y$

Differentiating w.r.t $t$ on both sides we get,

$\frac{dp}{dt}=2\frac{dx}{dt}+2\frac{dy}{dt}$

Now substituting the values for $\frac{dx}{dt}$ and $\frac{dy}{dt}$ we get,

$\frac{dP}{dt}=2(-3)+2\left(2\right)$

$=-2cm/min$

Hence the perimeter decreases at the rate of $-2cm/min$.

Area of the rectangle is $xy$

$A=xy$

Differentiate on both sides w.r.t $t$, we get

$\frac{d}{dt}\left(uv\right)=v\frac{du}{dt}+u\frac{dv}{dt}$

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

Substituting the values for $x,y,\frac{dy}{dt}$ and $\frac{dx}{dt}$ is

$\frac{dA}{dt}=10\times \left(2\right)+6\times (-3)$

$=20-18=2c{m}^2/min$

Hence the area of the rectangle increases at the rate of $2c{m}^2/min$