Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If the volume V of a sphere with radius r is V=(4/3)πr^3. If the surface area is s=4πr^2, how can I express the volume as a function of the surface area S? My first thought was to set them equal to each other but it doesn't seem like the right thing to do. Any hints/help would be appreciated

-edit- ok so from S=4πr^2, I got r=√(S/4π)...So now I am stuck as to what to do. Since I'm trying to do V(S)=?...I just replaced the r from V=(4/3)πr^3 with √(S/4π) but the answer doesn't correlate with the answer on the book. what am i doing wrong?

-edit-

V(S)=4/3π*(sqrt(S/4π)^3

V(S)=4/3π*((S^3/2)/(8π^3/2)

V(S)=(4Sπ^3/2)/(24π^3/2)

V(S)=(1Sπ^3/2)/(6π^3/2)

Answer should be:S/6*sqrt(S/π)...what did i do wrong?

Edit: V(S)=Sπ√π/6π√π

V(S)=S/6

Edit: V=4/3π*r^3

S=4π*r^2

r^2=S/4π

r=sqrt(S/4π)

V(S)=4/3π*(sqrt(S/4π)^3

V(S)=4/3π*S√S/8π√π

V(S)=S√S/3*2√π

V(S)=S√S/6√π

Special thanks to J.M for helping me figure this out

share|improve this question
    
The straight forward method involves solving for $r$ in one of the equations, replacing the $r$ in the other equation with the expression for $r$ you've just derived, and then solve for the volume. Can you follow this? –  J. M. Sep 21 '11 at 0:41
    
alright, I'm following you. so Since I want V(S): i should solve r for the S equation right? and then do the math? –  Ronnie.j Sep 21 '11 at 1:00
    
That works. :) If you figure out the answer, you can post your solution to answer your question. –  J. M. Sep 21 '11 at 1:02
    
can you look at my edits and see what i did wrong? –  Ronnie.j Sep 21 '11 at 1:10
    
You did try simplifying after replacing the $r$? Note also that $\sqrt{\frac{S}{4\pi}}$ is the same as $\frac12\sqrt{\frac{S}{\pi}}$. –  J. M. Sep 21 '11 at 1:14
show 12 more comments

1 Answer

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

$V=\frac{4}{3}(\sqrt{\frac{S}{4\pi}})^3\pi \Rightarrow V=\frac{4}{3}\frac{S}{4\pi}\pi\sqrt{\frac{S}{4\pi}} \Rightarrow V=\frac{S}{3}\sqrt{\frac{S}{4\pi}} \Rightarrow V=\frac{S}{3}\sqrt{\frac{1}{4}}\sqrt{\frac{S}{\pi}}\Rightarrow V=\frac{S}{6}\sqrt{\frac{S}{\pi}}$

share|improve this answer
    
Thanks! Very helpful –  Ronnie.j Sep 21 '11 at 20:31
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.