AMC 10 · 2023 · #17

Grade 8 geometry-3d
space-diagonal-formulasystems-of-equationssurface-areavolume-rectangular-prismvieta-formulas identify-subproblemsconvert-to-algebra ↑ Prerequisites: systems-of-equationssurface-area
📏 Medium solution 💡 2 insights
Problem
A rectangular box has distinct edge lengths a, b, c. The total edge length is 13, the total surface area is 112\frac{11}{2}, and the volume is 12\frac{1}{2}. Find the length of the longest interior (space) diagonal.

Pick an answer.

(A)
~2
(B)
$~\frac{3}{8}$
(C)
$~\frac{9}{8}$
(D)
$~\frac{9}{4}$
(E)
$~\frac{3}{2}$

AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

We never need the individual edges — only the symmetric sums s₁=a+b+c and s₂=ab+bc+ca. Tool #7 (Identify Subproblems) chops the job into three rungs: (a) read off s₁ and s₂ from the given totals, (b) plug into the identity (s₁)² = a²+b²+c² + 2s₂ to get a²+b²+c², (c) take the square root via the Pythagorean-theorem extension to find the space diagonal. Tool #13 (Convert to Algebra) handles the identity; Tool #8 (Analyze the Units) confirms the answer is a length (not an area or volume).

1STEP 1

Read the two symmetric sums off the totals: 4(a+b+c)=13 gives a+b+c=134\frac{13}{4}, and the six faces give 2(ab+bc+ca)=112\frac{11}{2}.

a+b+c=134\frac{13}{4}, 2(ab+bc+ca)=112\frac{11}{2}
2STEP 2

The identity (a+b+c)² = a²+b²+c² + 2(ab+bc+ca) gives a²+b²+c² = (134\frac{13}{4})² - 112\frac{11}{2} = 8116\frac{81}{16}; the volume 12\frac{1}{2} is a red herring.

a²+b²+c² = (a+b+c)² - 2(ab+bc+ca) = (134\frac{13}{4})² - 112\frac{11}{2} = 16916\frac{169}{16} - 8816\frac{88}{16} = 8116\frac{81}{16}
3STEP 3

The space diagonal is d=√(a²+b²+c²) by the 3D Pythagorean theorem; substituting 8116\frac{81}{16} gives d=√(8116\frac{81}{16})=94\frac{9}{4}.

d = √(8116\frac{81}{16}) = 94\frac{9}{4} → (D) 94\frac{9}{4}
4STEP 4

Units check: (a+b+c)² and 2(ab+bc+ca) are both areas, so their difference is an area, and its square root is a length — a diagonal.

[length]² - [length]² = [length]², √([length]²) = [length]
Answer
~94\frac{9}{4}
Sanity-check the magnitudes: a+b+c = 3.25 and a typical edge is ∼ 1, so the diagonal √(a²+b²+c²) should be a bit bigger than any single edge but smaller than a+b+c. The answer 94\frac{9}{4}=2.25 comfortably fits between ∼ 1 and 3.25. Also, (94\frac{9}{4})² = 8116\frac{81}{16} matches our intermediate value exactly. The volume abc=12\frac{1}{2} was never used — which is fine, because the diagonal is determined by s₁ and s₂ alone.
💡Key takeaway

You never need the individual edges a,b,c: the totals give a+b+c=134\frac{13}{4} and 2(ab+bc+ca)=112\frac{11}{2}, and the identity (a+b+c)² = a²+b²+c² + 2(ab+bc+ca) delivers a²+b²+c² = 8116\frac{81}{16}. The space diagonal is √(8116\frac{81}{16}) = (D) 94\frac{9}{4} — and the volume was a red herring.