Question

when an ice cube is put in warm water, it starts to melt.
in a particular case, the expression

v(t) = 8(1 - t)sqrt{1 - t}

is used to model the volume, ( v ), (in ( mathrm{cm}^3 )) ( t ) minutes after an ice cube is put in water.
find the initial rate of change of the volume of the ice cube.
give your answer as an integer. do not include units with your answer.

Explanation:

Step1: Simplify the function

First, rewrite \(V(t)=8(1 - t)\sqrt{1 - t}=8(1 - t)^{\frac{3}{2}}\).

Step2: Differentiate using the chain rule

Let \(u = 1 - t\), then \(V = 8u^{\frac{3}{2}}\).
The derivative of \(V\) with respect to \(u\) is \(V^\prime_u=8\times\frac{3}{2}u^{\frac{1}{2}} = 12u^{\frac{1}{2}}\), and the derivative of \(u\) with respect to \(t\) is \(u^\prime_t=-1\).
By the chain rule \(\frac{dV}{dt}=\frac{dV}{du}\times\frac{du}{dt}\), so \(\frac{dV}{dt}=12(1 - t)^{\frac{1}{2}}\times(-1)=- 12\sqrt{1 - t}\).

Step3: Find the initial rate of change

The initial time is \(t = 0\). Substitute \(t = 0\) into \(\frac{dV}{dt}\):
\(\frac{dV}{dt}\big|_{t = 0}=-12\sqrt{1-0}=-12\).

Answer:

\(-12\)