<PRE><B>Question 228828</B>
The volume of a cube with the side length x is V(x)=x^3.
 Find the rate of change of volume with respect to the side length if the surface area is 54cm^2.
:
We know the surface area of a cube = 6x^2 (total area of the six sides)
6x^2 = 54
x^2 = {{{54/6}}}
x^2 = 9
x = 3 
:
Vol = x^2
V = 3^3
V = 27 cu/cm 
:
We can say the relationship {{{SA/Vol}}} = {{{(6x^2)/x^3}}} = {{{6/x}}} (where x = 1 side)
:
:
See if this is true
Let v = 125 cu/cm
x^3 = 125
x = {{{3sqrt(125)}}}
x = 5 cm
:
Find the surface area
SA = 6(5^2)
SA = 6(25)
SA = 150
:
{{{150/125}}} = {{{6/5}}}
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