related rates of change

Question

A rectangular block of ice is melting at a rate of 2cm3/s. The block has a square base and its height is 3 times the base length. Find the rate at which the base length is changing when it is 8cm.

Answer 1

If the length of one side of the base is  s. Then the height is 3s.

V=area of base*height=s^2*(3s)=3s^3

So at time t, V(t)=3*s(t)^3, where V(t) and s(t) are volume and length of the base at time t.

Now differentiate with respect to t:

dV/dt=9s^2*(ds/dt) {by the chain rule}

2=9*(8^2)*ds/dt
ds/dt=2/(9*64)=1/288 cm/s