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| A set of interesting math problems which have simple and elegant solutions or proofs - a great resource for high school math classes and independent study! |
Rectangular Prism ProblemGiven a rectangular prism, the surface area is 94, and the sum of the lengths of the edges is 48. Find the length of an interior diagonal of the prism.SolutionThis problem is one of a very large class of interesting math problems in which it seems as though there isn't enough information to solve it. You have three unknowns (length, width, and height), but only two equations. How can you possibly find the lengths of the sides?The answer to that question is: you don't need to find the lengths of the sides, because that's not what the problem is asking for! It's possible to find the length of the interior diagonal without knowing the lengths of the sides. Let the dimensions of the prism be x, y, and z. 4(x + y + z) = 48, so (x + y + z) = 12 (Eq. #1). 2xy + 2yz + 2xz = 94 (Eq. #2). Now square (x + y + z) and you get: x2 + y2 + z2 + 2xy + 2yz + 2xz = 144 (Eq. #3). Subtract Eq. #2 from this and you get: x2 + y2 + z2 = 50. This is the square of the length of an interior diagonal, so the answer is: the square root of 50. (By the way, in creating this problem, I started with a prism with dimensions 3, 4, and 5. You can verify that it works, but you don't ever actually use those numbers in solving the problem.) Isn't that slick?
"Slick Math" is written by Douglas Twitchell, and hosted at The Problem Site. Contents copyright 2008 by Douglas Twitchell. Contents of this page may not be reproduced without permission of the author.
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