2023 AMC 10B Problem 17

Question

A rectangular box

\mathcal{P}

has distinct edge lengths

a,

b,

and

c.

The sum of the lengths of all

12

edges of

\mathcal{P}

is

13,

the sum of the areas of all

6

faces of

\mathcal{P}

is

\frac{11}{2},

and the volume of

\mathcal{P}

is

\frac{1}{2}.

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

\mathcal{P}?

2

\dfrac{3}{8}

\dfrac{9}{8}

\dfrac{9}{4}

\dfrac{3}{2}

The edge sum gives

a + b + c

and the face-area sum gives

ab + bc + ca

The diagonal is

\sqrt{a^2+b^2+c^2};

expand

(a+b+c)^2

to recover its radicand from the two known symmetric sums

The

12

edges give

4(a + b + c) = 13,

so

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

The

6

faces give

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

so

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

The space diagonal is

\begin{gathered} \sqrt{a^2 + b^2 + c^2} \\ = \small \sqrt{(a+b+c)^2 - 2(ab+bc+ca)} \\ = \sqrt{\frac{169}{16} - \frac{11}{2}} \\ = \sqrt{\frac{81}{16}} = \frac{9}{4}. \end{gathered}

Thus, D is the correct answer.

Problem 17 in other years