Optimizing Tin Can Dimensions for Minimum Surface Area

[rum2563]

[SOLVED] Optimization Problem

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

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 = \sqrt[3]{c/(4pi)}


After this I don't know what to do. Can someone please guide me? Thanks.

 

[G01]

You are trying to find the quantity, d/h for a given c, meaning you want to get d/h in terms of c. Now,

\frac{d}{h}=\frac{2r}{h}

Can you fill into the above equation and find d/r in terms of c?

 

[rum2563]

Thank you for response GO1.

If I understand you correctly, here is what I did:

d= diameter
h = height

So, as you said \frac{2r}{h}, I simply replace h with \frac{C}{pi(r^2)}

Now, I get \frac{2pi(\frac{c}{4pi})}{c} (the third root and cube cancel each other out in the step before this)

And then, \frac{d}{h}=\frac{1}{2}

Is that right? Please correct me If I am wrong. Thanks.

 

[G01]

I also end up with 1/2 as my answer. I think you got it.

 

[rum2563]

thanks very much. But are you 'sure' that I have done it right? Or do you 'think' I have done the question correctly? It would be helpful for a good confirmation.

Thanks for your help.

 

[G01]

I am quite sure you got the problem correct.

 

[rum2563]

thanks.

 

[rum2563]

hold on. can you please check again.

I think the surface area is: 2pir^2 + 2cr^-1

 

[G01]

hold on. can you please check again.

I think the surface area is: 2pir^2 + 2cr^-1

Indeed it is. Sorry about that. Your method to solve the problem is still correct though. Work the problem the same way but with the correct SA formula.

Again,I'm very sorry. Not that this is an excuse, but I usually help here in between my own work and classes and can get distracted while working on a problem. That was probably what happened here.

 

[HallsofIvy]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

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)

No, not "4hr- c/r^2", 4pi r- c/r^2.

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

Now, its pi again!

After this I don't know what to do. Can someone please guide me? Thanks.

You've found r and you know that h= c/(pi r^2). What is h? What is their ratio?

 

[rum2563]

G01, thanks for your help. I was going over the question when I realized that it had mistakes. But your help allowed me to follow the question and make sure to get it right. Now, I have got it, thanks for your help.

HallsofIvy, thanks to you too. You helped me fix the formula. It was great work. I actually didn't realize my mistakes after some long time...lol. But thanks a lot.

The answer I got is 1:1 for the ratio of height to diameter.