A hypothetical cube shrinks at a rate of 8 m³/min. At what rate are the sides of the cube changing when the sides are 3 m each?

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Answer 1 · 2016-10-12T13:33:06

When the sides are #3# #m# long, they are decreasing at a rate of #8/27# #m#/#min#.

Identify the Variables
The units #m^3#/#min# are units for volume over time. We are also asked about the sides of the cube. The variables are:

#V# = the volume of the cube

#x# = the length of a side of the cube

#t# = time in minutes

Identify the Rates of Change

The volume of the cube is decreasing at 8 #m^3#/#min#, so

#(dV)/dt = -8# #m^3#/#min#,.

We are asked to find the rate at which the sides are changing, so we want to

find #dx/dt# when #x = 3# #m#

Find an Equation Relating the Variables

The volume of a cube is given by the equation

#V = x^3#

Differentiate To find the equation relating the variables and their rates of change.

#(dV)/dt = 3x^2 dx/dt#

Plug in what you know and solve for what you're looking for.

#-8 =3 (3^2) dx/dt#

#27 dx/dt = -8#

#dx/dt = -8/27#

Answer the question

When the sides are #3# #m# long, they are decreasing at a rate of #8/27# #m#/#min#.

If you prefer to use units all the way through:

#-8 m^3/min=3 (3m)^2 dx/dt#

#27 m^2 dx/dt = -8 m^3/min#

#dx/dt = -8/27 m/min#