Question

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

Answer 1

Let length of rectangle $=xcm$

and width of rectangle $=ycm$

Given the length is decreasing at the rate of $3cm$/ minutes i.e. is decreasing w.r.t time

$\frac{dx}{dt}=-3cm/min.......\left(1\right)$ as $x$ is decreasing and the width $y$ is increasing at the rate of $2cm/min$

i.e. $y$ is increasing w.r.t time

$dy/dt=2cm/min.........\left(2\right)$

(1) Let $P$ be the perimeter of rectangle

$=2(l+width)$

$P=2(x+y)$

$dp/dt=2,(dx/dt+dy/dt)$

Given : $dx/dt=-3,dy/dt=2$

$\therefore dp/dt=2(-3+2)=2\times -1=-2cm/min$

$\therefore$ perimeter is decreasing at the rate of $2cm/min$

(2) Let $A$ be the area of the rectangle

$A=l\times width=xy\Rightarrow dA/dt=xdy/dt+y\frac{dx}{dt}$

$\Rightarrow dA/dt=-3y+2x$ from $\left(1\right)$ & $\left(2\right)$

$dA/dt{\frac{}{}|}_{x=10,y=6}=-3\times 6+2\times 10=-18+20=2$

Since are is $mc{m}^{2}dA/dt=2c{m}^2/min$

Hence are is increasing at the rate of $2c{m}^2/min$