2014 AMC 12B Problem 14

Below is the professionally curated solution for Problem 14 of the 2014 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 12B solutions, or check the answer key.

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Concepts:rectangular prism surface area algebraic manipulation

Difficulty rating: 1840

14.

A rectangular box has a total surface area of $94$ square inches. The sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals?

$8\sqrt{3}$

$10\sqrt{2}$

$16\sqrt{3}$

$20\sqrt{2}$

$40\sqrt{2}$

Solution:

Let the edges be $x, y, z.$ Then $xy+yz+zx = 47$ and $x+y+z = 12.$ Therefore $x^2+y^2+z^2 = (x+y+z)^2 - 2(xy+yz+zx) = 144 - 94 = 50.$

Each of the $4$ interior diagonals has length $\sqrt{x^2+y^2+z^2} = \sqrt{50} = 5\sqrt2,$ so their total length is $4 \cdot 5\sqrt2 = 20\sqrt2.$

Thus, the correct answer is D.

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