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?

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- Oct 24th 2006, 02:58 PMDragonVolume
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?

- Oct 24th 2006, 05:49 PMtopsquark
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