Boy do I have a fun one for you today!
A peice of sheet metal has 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
Lets call the height of the box and its width.
Then, you should know what is the volume of the box in function of and and, thanks to what you've done before, you can rewrite it as a function of only one variable. (say )
This gives a condition on the derivative of the volume which gives us and from which we get .that has the largest possible volume.