Question

The length of rectangle is decreasing at the rate of 5 cm / min and the width y is increasing at the rate of 4cm/min when x=8 and y=6 , find the rate of change of
(a) the perimeter and (b) the area of the rectangle?

Answer 1

Length $x$ decreasing $@5cm/min$

width $y$ increasing $=4cm/min$

a) Perimeter

$\frac{dy}{dx}=-5cm/min$ $P=2(l+b)$

$\frac{dy}{dt}=4cm/min$ $=2(x+y)$

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

$=-2cm/min$

b) Area of rectangle

Rate of change of area

$\frac{dy}{dt}=4cm/min\frac{dx}{dt}=-5cm/min$

$A=l\times b=x\times y$

$\frac{dA}{dt}=\frac{d}{dt}+x\frac{dy}{dt}$

$=[6\times (-5)+8\times (4\left)\right]c{m}^2/min$

$=[-30+32]c{m}^2/min$

$=2c{m}^2/min$