# Math

If 1,200 cm^2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter.

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1. You want a square base box with a fixed surface area:

SA = s² + 4sh
1200 = s² + 4sh

What's the maximum volume you can make with that box?

V = s²h

Solve the first equation for h, then substitute and solve for s into the volume equation:

1200 = s² + 4sh
-4sh = s² - 1200
h = -s/4 + 300/s

V = s²h
V = s²(-s/4 + 300/s)
V = -s³/4 + 300s

Now that we have volume in terms of one variable, we can find its maximum by taking the first derivative of that function, set it to zero, then solve for s:

dV/ds = -(3/4)s² + 300
0 = -(3/4)s² + 300
(3/4)s² = 300
s² = 400
s = 20

(throwing out the negative value of "s" since we're dealing with a box)

Now that we have "s", substitute it back into the volume equation to determine what that volume is:

V = -s³/4 + 300s
V = -(20)³/4 + 300(20)
V = -8000/4 + 6000
V = -2000 + 6000
V = 4000

The maximum volume is 4000 cm³ when the side of the box is 20cm² and the height is 10 cm.

V = s²h
4000 = 20²h
4000 = 400h
h = 10

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