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Question
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?

Answer
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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Janet Yang

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Word problems are my favorite type of math questions! I would not feel comfortable answering questions that require specialized knowledge (Physics, Statistics, etc.) because I have not studied these in depth.

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I tutor students (fifth through twelfth grades) and am a Top Contributor on Yahoo!Answers with over 24,000 math solutions.

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Co-author of An Outline of Scientific Writing: For Researchers With English as a Foreign Language.

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I have a Bachelor's degree in Applied Mathematics from the University of California at Berkeley, and a Master of Business Administration degree from The Wharton School.

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George White Elementary School. Homework Help program at the Ridgewood Public Library, Ridgewood, NJ. Individual students.

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