A rectangular box P has distinct edge lengths a, b, and c. The sum of the lengths of all 12 edges of P is 13, the sum of the areas of all 6 faces of P is (11)/(2), and the volume of P is (1)/(2). What is the length of the longest interior diagonal connecting two vertices of P?

2023 AMC 12B Problem 13 — Solution

2023 AMC 12B Problem 13

Below is the professionally curated solution for Problem 13 of the 2023 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12B solutions, or check the answer key.

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Concepts:rectangular prism algebraic manipulation

Difficulty rating: 1500

13.

A rectangular box $P$ has distinct edge lengths $a,$ $b,$ and $c.$ The sum of the lengths of all $12$ edges of $P$ is $13,$ the sum of the areas of all $6$ faces of $P$ is $\tfrac{11}{2},$ and the volume of $P$ is $\tfrac{1}{2}.$ What is the length of the longest interior diagonal connecting two vertices of $P?$

$2$

$\dfrac{3}{8}$

$\dfrac{9}{8}$

$\dfrac{9}{4}$

$\dfrac{3}{2}$

Solution:

From the edges, $4(a+b+c)=13,$ so $a+b+c=\tfrac{13}{4}.$ From the faces, $2(ab+bc+ca)=\tfrac{11}{2},$ so $ab+bc+ca=\tfrac{11}{4}.$ Then $a^2+b^2+c^2=\left(\tfrac{13}{4}\right)^2-2\cdot\tfrac{11}{4}=\tfrac{169}{16}-\tfrac{88}{16}=\tfrac{81}{16},$ so the diagonal is $\sqrt{\tfrac{81}{16}}=\tfrac{9}{4}.$

Thus, the correct answer is D.

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