# A simple mathematical problem

1. Sep 16, 2005

### vondutchgurl

i was given an assignment in my design engineering class......it goes as follows:

What is the maximum amount of weight that can float in a boat made out of 1 sheet of cardboard 4 feet by 8 feet in size...when flat but it can be folded and cut and glued in any shape you choose.

**you also have to be able to prove your solution**

if anyone could help me figure this out i would really appreciate it

thanks---
brooke

2. Sep 16, 2005

### vondutchgurl

if i did my calculation correct...then the maximum weight it could hold is 1010.31 pounds...im jst not sure how i would cut or fold the 8' by 4' sheet of cardboard to be able to prove my solution

3. Sep 16, 2005

### physmurf

I vote none since the boat will sink after a very short time after being introduced to the water.

4. Sep 16, 2005

### LeonhardEuler

If the cardboard is allowed to take any shape, then this is a very complicated problem involving the calculus of variations. If we are confining ourselves to box shapes, the problem becomes much more managable. Call the dimentions x, y, and z. You have to find the box of maximal volume with a given surface area. (This will be the box that holds the most weight since the amount of weight an object can have before sinking is the wieght that water of an equivalent volume has.) The thing to notice is that the top of the box does not need to be covered and shouldn't be in the box of maximal volume since it would be a waste of surface area. If x is the length, y is the hieght, and z is the other dimention, then one of the xz surfaces does not need to be there. So you must maximize
$$f(x,y,z)=xyz$$
subject to
$$2xy + 2yz + xz=32$$
The standard way of doing this would be the method of Lagrange multipliers.