Suppose that eh volume of a ball is equal to Pie/6 times the volume of a cube What is the ration of the surface area of the ball to that of the cube?
I must be in a crabby mood tonight.
Pet peeve warning!!
It's spelled "pi" not "pie."
Pet peeve over.
Okay. We know that $\displaystyle V_b = \frac{ \pi }{6} \cdot V_c$.
We also know that
$\displaystyle V_b = \frac{4}{3} \pi r^3$
$\displaystyle V_c = s^3$
$\displaystyle S_b = 4 \pi r^2$
$\displaystyle S_c = 6s^2$
(Where "s" is the side of the cube.)
So, solve $\displaystyle V_b$ for r:
$\displaystyle r = \left ( \frac{3}{4 \pi}V_b \right ) ^{1/3}$
So
$\displaystyle S_b = 4 \pi \left ( \frac{3}{4 \pi}V_b \right ) ^{2/3}$
Sim: $\displaystyle S_c = 6 \left ( V_c \right ) ^{2/3}$
Now sub in the relation for $\displaystyle V_b$ into the $\displaystyle S_b$ equation and take the ratio.
-Dan