AMC 10 · 2021 · #14

Grade 10 geometry-3d

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

volume-pyramidpythagorean-theoremdifference-of-squaressystems-of-equations convert-to-algebraspatial-visualizationsystematic-enumeration ↑ Prerequisites: pythagorean-theoremvolume-pyramid

📏 Long solution 💡 4 insights

Problem

A rectangle lies flat, and a segment rises from one corner straight out of its plane. That segment's length is a whole number, and the three slant distances from its top to three corners are consecutive odd positive integers in order. Find the volume of the pyramid whose apex is that top and whose base is the rectangle.

Pick an answer.

(A)

$24\sqrt5$

(B)

(C)

$28\sqrt5$

(D)

(E)

$8\sqrt{70}$

How to solve

Strategy Introduce a Variable

The volume of a pyramid needs two things: the area of the base and the height. Here the height is DM and the base area is DA times DC, so the whole problem reduces to finding three lengths. Nothing is numerical yet, so the first move is to name those three lengths and name the smallest of the three consecutive odd integers. Because DM is perpendicular to the whole base plane, every one of MA, MC, MB is the hypotenuse of a right triangle that has DM as one leg. That gives three Pythagoras equations in the same letters. Two of the unknowns can then be swept out algebraically, leaving one equation that mixes an integer height with an integer slant distance, and the integer condition is exactly what pins the answer down.

1STEP 1

See the three right triangles

Standing perpendicular creates three right angles.

DM ⊥ plane ABCD → ∠ MDA = ∠ MDC = ∠ MDB = 90°

2STEP 2

Write Pythagoras three times

Three equations come out.

n² = h² + x², (n+2)² = h² + y², (n+4)² = h² + x² + y²

3STEP 3

Sweep out the base sides

Both base sides cancel.

(n+4)² = h² + (n² - h²) + ((n+2)² - h²) → h² = n² - 4n - 12

4STEP 4

Rewrite as a difference of squares

What remains is a difference of squares.

h² = (n-2)² - 16 → (n-2)² - h² = 16 → (n-2-h)(n-2+h) = 16

5STEP 5

Test the integer factor pairs

Integrality leaves one factor pair.

n-2-h = 2 ; n-2+h = 8 → n = 7, h = 3 → MA = 7, MC = 9, MB = 11, DM = 3

6STEP 6

Recover the rectangle's sides

Recover both side lengths.

x² = 7² - 3² = 40, y² = 9² - 3² = 72 → DA = 2√(10), DC = 6√(2)

7STEP 7

Apply the volume formula

The volume is twenty-four root five.

V = 1/3 (DA · DC) (DM) = 1/3(2√(10) · 6√(2))(3) = 12√(20) = 24√(5)

Answer

24√5

Every condition of the problem holds for the numbers found. DM = 3 is a positive integer. MA = 7, MC = 9, MB = 11 are consecutive odd positive integers in that order. The three Pythagoras equations check out: 9 + 40 = 49, 9 + 72 = 81, and 9 + 40 + 72 = 121. Numerically the base is about 6.32 by 8.49, so its area is about 53.7, and multiplying by the height 3 and by 1/3 leaves about 53.7, which agrees with 24 times the square root of 5, about 53.67. The other choices are all larger: 60, 66, 28 times the square root of 5 is about 62.6, and 8 times the square root of 70 is about 66.9. A base of roughly 6 by 8 with height 3 cannot reach those, so 24 times the square root of 5 is the only reasonable size.

💡Key takeaway

When several slant distances all lean on the same vertical leg, write a Pythagoras equation for each one, subtract to make the unwanted lengths disappear, and let the leftover difference of squares plus the whole-number condition finish the job.

See the three right triangles
Name the lengths and write Pythagoras three times
Sweep out the base sides x and y
Rewrite it as a difference of squares
Test the integer factor pairs
Recover the sides of the rectangle
Apply the pyramid volume formula