Calculus Urgently Needed: Finding height to diameter ratio

[truthfulone]

A tin can is to have a given capacity. Find the ratio of height to diameter if the amount of tin (total surface area) is a minimum.

c=pi(r^2)h
surface area = 2pi(r^2)+2h(pi)r

h= c/(pi(r^2))

surface area = 2pi(r^2)+2h(pi)r
= 2pi(r^2)+2pi(r)(c/(pi(r^2)))
= 2pi(r^2) + c(r^-1)

Now, derivate of surface area:

SA` = 4hr - c/(r^2)
0 = ( 4h(r^3)-c )/(r^2)
4h(r^3)=c
r = \(\displaystyle \sqrt[3]{c/(4pi)}\)

Am I on the right track? Also, after finding r, what should I do? Please help. I really need this urgently. Thanks.

 

[galactus]

As you obviously know, the volume is given by \(\displaystyle V={\pi}r^{2}h\)..[1]

Surface area by \(\displaystyle S=2{\pi}rh+2{\pi}r^{2}\)...[2]

Solve [1] for h ad sub into [2]:

\(\displaystyle h=\frac{V}{{\pi}r^{2}}\)

\(\displaystyle S=2{\pi}r(\frac{V}{{\pi}r^{2}})+2{\pi}r^{2}=\frac{2V}{r}+2{\pi}r^{2}\)

\(\displaystyle \frac{dS}{dr}=4{\pi}r-\frac{2V}{r^{2}}\)

Set to 0 and solve for r, we find \(\displaystyle r=\frac{2^{\frac{2}{3}}\cdot{V^{\frac{1}{3}}}}{2{\pi}^{\frac{1}{3}}}\)

So, the diameter is twice that/

Sub this into the equation for h to find h. Then you can find the ratio of height to diameter.

If you do it correctly, you will find the minimum is achieved when the diameter and height are the same.

 

[truthfulone]

wow. Thanks for your help. The tex makes it easier to read those equations. Great help. I actually got 1:1 ratio for height to diameter. You are great. Thanks.