1. ## difficult indices question

The Volume and surface area of a sphere is 4/3πr^3 and 4πr^2 respectively. V=4/3πr^3 and S=4πr^2. Write a) S in terms of V and b) V in terms of S

I'm stuck on this question... I write out similar base units and stuff but it doesn't seem to work, any help on the steps I should take to tackle this question?

The answer to part a) -
S=2^2/3 3^2/3 π^1/3 V^2/3
I know the answer as its written on the book but I don't know the steps involved

Thanks.

2. Originally Posted by david18
The Volume and surface area of a sphere is 4/3πr^3 and 4πr^2 respectively. V=4/3πr^3 and S=4πr^2. Write a) S in terms of V and b) V in terms of S

I'm stuck on this question... I write out similar base units and stuff but it doesn't seem to work, any help on the steps I should take to tackle this question?

The answer to part a) -
S=2^2/3 3^2/3 π^1/3 V^2/3
I know the answer as its written on the book but I don't know the steps involved

Thanks.
$\displaystyle V = \frac{4}{3} \pi r^3$
and
$\displaystyle S = 4 \pi r^2$

The only commonality between these is r. (Well, there's a $\displaystyle \pi$, too , if you want to be technical.) So solve one of your equations for r:
$\displaystyle S = 4 \pi r^2$

$\displaystyle r^2 = \frac{S}{4 \pi}$

$\displaystyle r = \sqrt{\frac{S}{4 \pi}}$

Thus
$\displaystyle V = \frac{4}{3} \pi \left ( \sqrt{\frac{S}{4 \pi}} \right ) ^3$

Just because I have a love/hate thing going for radicals at the moment, I'm going to square both sides of this to get rid of it:
$\displaystyle V^2 = \left ( \frac{4}{3} \pi \right ) ^2 \left ( \frac{S}{4 \pi} \right ) ^3$

$\displaystyle V^2 = \frac{16}{9} \pi ^2 \cdot \frac{S^3}{64 \pi ^3}$

$\displaystyle V^2 = \frac{S^3}{36 \pi }$

Can you take things from here?

-Dan

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