2023 AMC 12B Problem 13

Problem 13 of 25IntermediateAlgebra Geometry

A rectangular box

P

has distinct edge lengths

a,

b,

and

c.

The sum of the lengths of all

12

edges of

P

13,

the sum of the areas of all

6

faces of

P

\tfrac{11}{2},

and the volume of

P

\tfrac{1}{2}.

What is the length of the longest interior diagonal connecting two vertices of

P?

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Solution

From the edges,

4(a+b+c)=13,

a+b+c=\tfrac{13}{4}.

From the faces,

2(ab+bc+ca)=\tfrac{11}{2},

ab+bc+ca=\tfrac{11}{4}.

Then

\begin{aligned} a^2+b^2+c^2 &=\left(\tfrac{13}{4}\right)^2-2\cdot\tfrac{11}{4} \\ &=\tfrac{169}{16}-\tfrac{88}{16} \\ &=\tfrac{81}{16}, \end{aligned}

so the diagonal is

\sqrt{\tfrac{81}{16}}=\tfrac{9}{4}.

Thus, the correct answer is D.