calculus question

Question

Each side of a square is increasing at a rate of 4 cm/s. At what rate is the area of the square increasing when the area of the square is 81 cm2?

Answer 1

rate of change of length per unit time:

dl/dt = 4

area of a square is given by:

a = l*l = l²

differentiating, we find rate of change of area per unit time:

da/dt = 2l*dl/dt

notice when a=81, l=9 (a=81=l², l=sqrt(81)=9)

therefore, da/dt when a=81 is given by:

da/dt = 2l*dl/dt = 2(9cm)*(4cm/s)

da/dt = 72 cm²/s

Answer 2

Let s = the side length of a square

Let A be the area of that square.

Then A(s) = s²

Taking the derivative of both sides with respect to "t" we get:

dA/dt = 2s(ds/dt) [this is an application of the chain rule]

We know ds/dt = 4 cm/s

We know A = 81 which means that s = 9

So using dA/dt = 2s(ds/dt)

dA/dt = 2(9)(4) = 72 cm²/s