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f1player

Junior Member
Joined
Feb 25, 2005
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Here's a question I'm having trouble with:

A cylindrical container without a top has a surface area of 3pi metres squared. What will be the height and radius if the container is to have the maximum capacity?

This is what I have done:

Surface Area = pi * r^2 + 2pi * r * h

3pi = pi * r^2 + 2pi * r * h

Therefore: h = (3pi - pi * r^2)/ (2pi* r)

Now Volume = pi * r^2 * h

Therefore subbing in for h: V = pi * r^2 * (3pi - pi * r^2)/(2pi * r)

At this point I need to differentiate to find the stationary value, but this expression is too complicated. Can it be simplified??
 
f1player said:
Here's a question I'm having trouble with:

A cylindrical container without a top has a surface area of 3pi metres squared. What will be the height and radius if the container is to have the maximum capacity?

This is what I have done:

Surface Area = pi * r^2 + 2pi * r * h

3pi = pi * r^2 + 2pi * r * h

Therefore: h = (3pi - pi * r^2)/ (2pi* r)You're just having

algebra trouble. Simplify here

\(\displaystyle \L\\h=\frac{3{\pi}-{\pi}r^{2}}{2{\pi}r}=\frac{3{\pi}}{2{\pi}r}-\frac

{{\pi}r^{2}}{2{\pi}r}\)

Cancellation:

\(\displaystyle \L\\h=\frac{3\sout{\pi}}{2\sout{\pi}r}-\frac{\sout{\pi}r^{\sout{2}}}{2\sout{\pi}\sout{r}}=\frac{3}{2r}-\frac{r}{2}\)

Now Volume = pi * r^2 * hYep

Therefore subbing in for h: \(\displaystyle \L\\V={\pi}r^{2}(\frac{3}{2r}-\frac{r}{2})=\frac{3{\pi}r}{2}-\frac{{\pi}r^{3}}{2}\)

Differentiate, set to 0 and solve for r.



At this point I need to differentiate to find the stationary value, but this expression is too complicated. Can it be simplified??
Click to expand...
 
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