1996 AMC 12 Problem 23

Attempt Problem 23 of the 1996 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1996 AMC 12 solutions, or check the answer key.

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23.

The sum of the lengths of the twelve edges of a rectangular box is

140,

and the distance from one corner of the box to the farthest corner is

21.

The total surface area of the box is

776

784

798

800

812

If the side lengths are

a,b,c

, translate the edge sum and space diagonal into equations

Expand

(a+b+c)^2

to obtain

2(ab+bc+ca)

directly

The edge sum gives

4(a+b+c)=140,

a+b+c=35.

The space diagonal gives

a^2+b^2+c^2=21^2=441.

Therefore the surface area is

\begin{gathered} 2(ab+bc+ca)\\ {}=(a+b+c)^2\\ {}-(a^2+b^2+c^2)\\ {}=35^2-441=784. \end{gathered}

Thus the correct answer is B.

Problem 23 in other years