# 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?

##### 1 Answer

Aug 20, 2016

The volume is decreasing at a rate of

#### Explanation:

Let the length of the cube be denoted as

The volume of the cube,

#V = l^3#

Since the sides are decreasing at a rate of

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

where *decreasing* with time.

To find the rate at which the volume change,

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

From simple differentiation, we know that

#frac{"d"V}{"d"l} = frac{"d"}{"d"l}(l^3) = 3l^2#

Therefore,

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

#= 3 xx (2"mm")^2 xx (-2"mm/s")#

#= -24 "mm"^3"/s"#

The volume is changing at a rate of