Question

An ice cube is melting, and the lengths of its sides are decreasing at a rate of 0.8 millimeters per minute. At what rate is the volume of the ice cube decreasing when the lengths of the sides of the cube are equal to 14 millimeters? Give your answer correct to the nearest cubic millimeter per minute.

Answer 1

Answer:

$\dot{V}=0.512\,m^3.min^{-1}$

Explanation:

rate of melting of the sides of ice cube, $\dot{L}=0.8\,mm.min^{-1}$

Since the cube is melting equally from all the three dimensions, therefore the rate of decrease of volume of the cube:

$\dot{V}=0.8\times 0.8\times 0.8=0.512\,m^3.min^{-1}$

The rate of melting is constant irrespective of the dimensions of the ice cube.

Answer 2

The volume of the cube is decreasing at a rate of -470.4 mm³/min

Rate of change:

Let the length of the sides of the cube be L. The rate of change sides is given as:

$\frac{dL}{dt}=-0.8\;mm/min$

the negative sign indicates that the length is decreasing.

Now, the volume of the cube is given by the following relation:

V = L³

If we take the time derivative of the above equation, then we can calculate the rate of change of its volume. So,

$\frac{dV}{dt}=3L^2\frac{dL}{dt}$

when the lengths of the sides of the cube are equal to 14 mm, the rate of change of volume is:

$\frac{dV}{dt}=3\times14^2\times\frac{dL}{dt}\\\\\frac{dV}{dt}=3\times14^2\times(-0.8)\\\\\frac{dV}{dt}=-470.4\;mm^3/mim$

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