1. ## Optimizing Equations

Hi everyone.
Boy do I have a fun one for you today!

A peice of sheet metal has $\displaystyle 4800 cm^2$ area.
Create an open top box with a quare bottom that has the largest
possible volume. Did not get very far on this one. I have no idea how
to solve this problem. :P

2. Hello,

Originally Posted by ffezz
Hi everyone.
Boy do I have a fun one for you today!

A peice of sheet metal has $\displaystyle 4800 cm^2$ area.
Create an open top box with a quare bottom that has the largest
possible volume. Did not get very far on this one. I have no idea how
to solve this problem. :P
Is there an information about the shape of the box, or the dimensions of the square bottom ?

3. Hi
Originally Posted by Moo
Is there an information about the shape of the box, or the dimensions of the square bottom ?
If it were the case, I think the problem couldn't be solved, there would be too many conditions. (unless it is useless information )

Lets call $\displaystyle h$ the height of the box and $\displaystyle x$ its width.
Originally Posted by ffezz
A peice of sheet metal has $\displaystyle 4800 cm^2$ area.
This gives a relation between $\displaystyle x$ and $\displaystyle h$ (the area of the box is the sum of the area of each side and has to be $\displaystyle 4800\mathrm{cm}^2$)

Then, you should know what is the volume of the box in function of $\displaystyle x$ and $\displaystyle h$ and, thanks to what you've done before, you can rewrite it as a function of only one variable. (say $\displaystyle h$)

that has the largest possible volume.
This gives a condition on the derivative of the volume which gives us $\displaystyle h$ and from which we get $\displaystyle x$.