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?
1 Answer
When the sides are
Explanation:
Identify the Variables
The units
#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
#(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
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#