Question

Each side of a square is increasing at a rate of 2 cm/s. at what rate is the area of the square increasing when the area of the square is 9 cm2?

Answer 1

Let x(t) =  the length of a side of the square (cm) at time t (s).
The rate of change of x is given as
$\frac{dx}{dt} =2 \, \frac{cm}{s}$

The area (cm²) at time t is 
A = x²

The rate of change of the area with respect to time is
$\frac{dA}{dt} = \frac{dA}{dx} \frac{dx}{dt} =2 \frac{dA}{dx} =2(2x)=4x \, \frac{cm^{2}}{s}$

When A = 9 cm², then x = √9 = 3cm. Hence obtain
$\frac{dA}{dt} = 4(3) = 12 \, \frac{cm^{2}}{s}$

Answer: 12 cm²/s