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I have a tricky word problem I just can't get. Thank you!

You are planning to make an open rectangular box from a 10 by 20 inch piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of the volume you can make this way, and what is its volume?

Let the cut-outs be x inches by x inches.

https://www.flickr.com/photos/dwread/16052356767/

The box will be (20-2x) inches long, (10-2x) inches wide, and x inches high.

https://www.flickr.com/photos/dwread/16052356817/

https://www.flickr.com/photos/dwread/16052356907/

volume V = length·width·height = (10-2x)(20-2x)x = 4x³-60x²+200x in³

At maximum volume, the first derivative is zero.

dV/dx = 0

12x²-120x+200 = 0

By the quadratic formula,

x = [120 ± √(120² – 4·12·200)] / [2·12]

= [120 ± √4800] / 24

= [120 ± 40√3] / 24

= 5 ± (1⅓)√3

≅ 2.11, 7.89

The cardboard is not wide enough to cut out 7.89-inch squares, so 7.89 is an extraneous solution.

x ≅ 2.11 inches

Dimensions of box:

10-2x = 5.77 inches wide

20-2x = 15.77 inches long

2.11 inches high

volume = 5.77·15.77·2.11 = 192.45 in³

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