# A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of 2 mm/sec. Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are 2 mm each?

Aug 20, 2016

The volume is decreasing at a rate of $24 \text{mm"^3"/s}$.

#### Explanation:

Let the length of the cube be denoted as $l$.

The volume of the cube, $V$, is given by

$V = {l}^{3}$

Since the sides are decreasing at a rate of $2 \text{mm/s}$, we write

frac{"d"l}{"d"t} = -2"mm/s",

where $t$ represents time. The negative sign is there as $l$ is decreasing with time.

To find the rate at which the volume change, frac{"d"V}{"d"t}, we can use the chain rule

frac{"d"V}{"d"t} = frac{"d"V}{"d"l} frac{"d"l}{"d"t}

From simple differentiation, we know that

$\frac{\text{d"V}{"d"l} = frac{"d"}{"d} l}{{l}^{3}} = 3 {l}^{2}$

Therefore,

frac{"d"V}{"d"t} = 3l^2 frac{"d"l}{"d"t}

$= 3 \times \left(2 \text{mm")^2 xx (-2"mm/s}\right)$

$= - 24 \text{mm"^3"/s}$

The volume is changing at a rate of $- 24 \text{mm"^3"/s}$.