Box Sizes

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In the Old West an express company delivers two boxes. The driver is to charge $5 per cubical foot as per instructions on the bill. The customer says the custom is to pay per running foot. The driver measured the length of the boxes and charges $5 per running foot. The boxes are square and one is half the height of the other. The driver placed the boxes together, measured the combined length and discovered less than .001 cent difference in either $5 per cubical foot or $5 per running foot.

Calculate sizes of the boxes. Thank you.

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@Randy Dandy,

The smaller cube has sides of 1/√3 feet·

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@ekename,

Thank you ekename. I have a question- The "answer" given to the problem is in square inches, square feet, and cubic feet. If the boxes could be any length and width (I am guessing), how could the specific numbers given be the "answer"? Thanks again.

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@Randy Dandy,

Let K = the length of the small box in feet

The length of the big box is therefore 2K feet

The cost of charging by volume is 5 x ( K^3 + [2K]^3 )
The cost of charging by running feet is 5 x ( K + 2K)

The cost is the same which ever method is used so:

5 x ( K^3 + [2K]^3 ) = 5 x ( K + 2K)

5 x (K^3 + 8K^3 ) = 5K + 10K

45K^3 = 15K and simplifying

3K^3 - K = 0

K ( 3K^2 -1) = 0

K = 1/ ROOT 3 = 1/ 1.7321 = 0.5773FEET = 0.5773 X 12 INCHES = 6.928INCHES

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@ekename,

Thank you ekename.

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