# Applied Maxima and Minima

1. Jan 18, 2005

Hello all

You are planning to make an open box from a 30- by 42 inch piece of sheet metal by cutting congruent squares from the corners and folding up the sides. You want the box to have the largest possible volume.

(a) What size square should you cut from each corner? (gice side length of square)

(b) What is the largest possible volume the box will have

I know $$V(x) = x(30-2x)(42-2x)$$

So $$V'(x) = (30 - 2 x) (42 - 2 x) - 2 x (42 - 2 x) - 2 x (30 - 2 x)$$

I find the critical points, however how do I find the maximum volume. Also I am not sure how you would find what square size you should cut. Shouldn't you find the maximum volume?

Thanks

2. Jan 18, 2005

### MathStudent

Both questions are really solved simultaneously. You are trying to find the value of x that makes V(x) take on the greatest value, thus that value for x will be the length of the side of the square you cut.

Here are some questions to get you started:
How do you find local extrema of a function?
How do you find absloute max?
what values of x make V(x) meaningless?