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
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