2014 AMC 12B Problem 14

Question

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}

Let the edges be

x, y, z,

so

2(xy+yz+zx)=94

and

4(x+y+z)=48

Use

(x+y+z)^2 =

x^2+y^2+z^2

+ 2(xy+yz+zx)

to find each diagonal

\sqrt{x^2+y^2+z^2}

Let the edges be

x, y, z.

Then

xy+yz+zx = 47

and

x+y+z = 12.

Therefore

\begin{gathered} x^2+y^2+z^2 = (x+y+z)^2 \\ {}- 2(xy+yz+zx) \\ = 144 - 94 \\ = 50. \end{gathered}

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.

Problem 14 in other years