## anonymous 5 years ago A box with an open top is to be constructed out of a rectangular piece of cardboard with dimensions length=8 ft and width=9 ft by cutting a square piece out of each corner and turning the sides up. Determine the length x of each side of the square that should be cut which would maximize the volume of the box.

Let x equal the dimension of the corner that is going to be cut out (so the sides fold up and make a box. This x will also equal the height of the box. The dimensions of the box will then be Length=8-2x (corners cut out at each end.) width =9-2x. Volume will equal l*w*h. Please review this and see if you follow with understanding.

$V=(9-2x)(8-2x) x$

3. anonymous

Following you so far. haha, that's about as far as I got on my own though. lol

$V=(72-34X+4X ^{2})x$

$V=72x-34x ^{2}+4x ^{3}$

Rearranging in to a more standard form$V=4x ^{3}-34x ^{2}+72x$ Are you still with me?

7. anonymous

Yep. Then we take the derivative?

Yes and set to zero and then solve for x.

Let me know what you get.

10. anonymous

Derivative of.. $12x ^{2} -68x +72$ ?

Yes, to make it easier set it to zero and divide both sides by 4 getting:

$3x ^{2}-17x+18=0$

13. anonymous

so I'm gettinggg.. $1/6 (17\pm \sqrt{73}$

14. anonymous

with another ) at the end.. lol

You're ahead of me. I was trying to factor before going for the quadratic equation. It looks like that was the way to go. I'll check it out and if there is an error I will be back. Remember this is the dimension of the corner cut out, to get the dimensions of the box you will subtract twice this from the 8' and 9'

16. anonymous

How do I know which x value to use though, the plus gives me approximately 1.4 and the minus gives me approximately 4.3

17. anonymous

Nevermind I've got it. haha thank you!