AMC 10 · 2014 · #14

Grade 9 geometry-3d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Symmetric Function of Roots via Vieta

surface-areaspace-diagonal-formulasymmetric-polynomialspythagorean-theorem convert-to-algebraspatial-visualization ↑ Prerequisites: surface-areapythagorean-theorem

📏 Medium solution 💡 3 insights

Problem

A box's surface area and total edge length are both known. Find the total length of its inner diagonals.

Pick an answer.

(A)

$8\sqrt{3}$

(B)

$10\sqrt{2}$

(C)

$16\sqrt{3}$

(D)

$20\sqrt{2}$

(E)

$40\sqrt{2}$

How to solve

Strategy Introduce a Variable

Name the edge lengths x, y, z (Tool #4) and the two clues become two equations. But those equations do not determine x, y, z individually — two equations, three unknowns — so any plan that starts "solve for the sides" is doomed. The way out is Tool #15 (Organize Information in More Ways): notice that the target quantity, like both clues, depends only on the symmetric combinations x+y+z and xy+yz+zx, and that squaring the first clue produces the second clue as a term inside it. Tool #17 (Visualize Spatial Relationships) is needed first, to settle what an interior diagonal even is: how many the box has and whether they all have the same length — the reference argument assumes this rather than showing it. Tool #7 splits the surface into face pairs, Tool #6 builds one concrete box to confirm the data is not self-contradictory, and Tool #3 supplies an independent check against the choices.

1STEP 1

Name the three edge lengths

The edge total gives the sum of the three.

4(x+y+z) = 48 → x+y+z = 12

2STEP 2

Split the surface into face pairs

The surface area gives their pairwise products.

2(xy+yz+zx) = 94 → xy+yz+zx = 47

3STEP 3

Count the interior diagonals

There are exactly four inner diagonals.

C(8, 2) = 28 = 12_edges + 12_face diagonals + 4_interior, total = 4√(x²+y²+z²)

4STEP 4

Square the edge sum

Squaring the edge sum links the two givens.

144 = (x+y+z)² = x²+y²+z² + 2(xy+yz+zx) = x²+y²+z² + 94 → x²+y²+z² = 50

5STEP 5

Confirm such a box exists

Such a box really exists.

3x²-24x+47=0 → x = 4 ± √(3)/3; (4+√(3)/3)² · 2 + (4-2√(3)/3)² = 50

6STEP 6

Add the four equal diagonals

Adding gives 20√2, choice (D).

4√(50) = 4 · 5√(2) = 20√(2) → (D)

Answer

20√(2)

Test the answer on the concrete box x = y ≈ 4.577, z ≈ 2.845 built in the fifth step: its edges total 4(12) = 48 inches, its surface area is 2(20.95+13.02+13.02) = 94 square inches, and its diagonal is √(4.577²+4.577²+2.845²) = √(50) ≈ 7.071 inches, so the four diagonals total about 28.28 inches — and 20√(2) ≈ 28.284. A second sanity test: a cube with the same edge sum has side 4 and diagonals 4√(3) ≈ 6.93, totalling 16√(3) ≈ 27.71. Our box is slightly less even than that cube, and with x+y+z held fixed, spreading the sides apart increases x²+y²+z², so the answer should be a little larger than 16√(3) — and 20√(2) is, by about half an inch. Both checks agree.

💡Key takeaway

You never need the three side lengths: squaring x+y+z makes the surface area appear inside it, and what is left over is exactly the square of the diagonal.

Name the three edge lengths
Split the surface into face pairs
Count the interior diagonals
Square the edge sum
Confirm such a box exists
Add the four equal diagonals