AMC 10 · 2021 · #9

Grade 11 geometry-3dalgebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

logarithm-propertiessurface-areavolume-rectangular-prismexponents convert-to-algebraidentify-subproblems ↑ Prerequisites: logarithm-propertiesvolume-rectangular-prism

📏 Medium solution 💡 3 insights

Problem

A box has its three edge lengths given as three logarithms of the same number, taken in bases 2, 3, and 4. The number measuring its surface area comes out the same as the number measuring its volume. Find that number.

Pick an answer.

(A)

$2\sqrt{6}$

(B)

$6\sqrt{6}$

(C)

(D)

(E)

576

How to solve

Strategy Introduce a Variable

The logarithms make the box look exotic, but a box does not care what its edges are named. Call the edges a, b, c and the geometry becomes routine: surface area 2(ab+bc+ca), volume abc. Setting those equal gives one messy-looking equation, and the single re-organizing move of dividing by abc turns it into a sum of reciprocals. Reciprocals are exactly the form logarithms handle best, because one over log₂x is just log_x2. Only at the very end do you work backwards from the logarithm to x itself.

1STEP 1

Rename the three edges

Give the three edges names.

a=log₂x, b=log₃x, c=log₄x, a,b,c ＞ 0

2STEP 2

Write surface area and volume

Write surface area and volume as formulas.

S=2(ab+bc+ca), V=abc

3STEP 3

Divide the equation by the volume

Dividing gives a sum of reciprocals.

2(ab+bc+ca)=abc ⟹ (2(ab+bc+ca))/abc=1 ⟹ 1/a+1/b+1/c=1/2

4STEP 4

Flip each logarithm upside down

Flipping each logarithm swaps its base.

1/a=1/log₂x=log_x2, 1/b=log_x3, 1/c=log_x4

5STEP 5

Merge three logs into one

Three logarithms merge into one.

log_x2+log_x3+log_x4=log_x(2 · 3 · 4)=log_x24=1/2

6STEP 6

Undo the logarithm to reach x

Undoing the logarithm gives 576.

log_x24=1/2 ⟹ x¹/2=24 ⟹ x=24²=576

Answer

576

Check x=576 exactly rather than with a calculator. Since 576=24², the identity log₅₇₆₂₄=1/2 holds on the nose, so 1/a+1/b+1/c=1/2 and multiplying back by abc restores 2(ab+bc+ca)=abc. The edges are also legal: 576 ＞ 1, so log₂₅₇₆≈ 9.17, log₃₅₇₆≈ 5.79, and log₄₅₇₆≈ 4.58 are all positive. Numerically the volume is about 9.17 × 5.79 × 4.58≈ 243 and the surface area is about 2(53.1+26.5+42.0)≈ 243, so the two really do land on the same number. It also makes sense that x has to be large: the edges grow only like logarithms, so x must be pushed far out before a box gets big enough for volume to catch up with surface area.

💡Key takeaway

When a box has surface area equal to volume, divide the equation by the volume: it collapses into 1/a+1/b+1/c=1/2, and the reciprocal of a logarithm is just that logarithm with its base and its number swapped.

Rename the three edges
Write surface area and volume
Divide the equation by the volume
Flip each logarithm upside down
Merge three logs into one
Undo the logarithm to reach x