SOLUTION: Suppose you wish to reduce the volume of a sphere by a factor of 7.00. If it's original radius was R, what is the reduced radius in terms of R? By what factor does the sphere's sur

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The formula for the volume of a sphere is
(1)
Making the sphere seven times smaller means
multiplying by
Let = the new radius when
is of what it used to be
(2)
Now I need to find in terms of
Multiply both sides of (1) by
(2)
(2)
Now I need to substitute in (1) for in (2)
(2)
(2)
Now take the cube root of both sides
(2)
is the reduced radius in terms of
The surface areas are proportional to and
Because
so I can rewrite (2) as
(2)
(2)
Now square both sides


This is the factor that the surface area changes by