AMC 10 · 2009 · #11

Grade 7 geometry-3d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
volume-rectangular-prismdifference-of-squareslinear-equations-one-var convert-to-algebra ↑ Prerequisites: exponents
📏 Medium solution 💡 2 insights
Problem
A cube has all three edges the same length. Build a new box from it: make one edge longer by 1, make another edge shorter by 1, and leave the third edge alone. The new box holds 5 less volume than the cube. Find the cube's volume.

Pick an answer.

(A)
8
(B)
27
(C)
64
(D)
125
(E)
216

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The cube's edge length is the one thing that ties both solids together, but it is unknown, so tool #4 (Introduce a Variable) names it s and lets us describe the cube and the new box with the same symbol. Tool #13 (Convert to Algebra) then turns the sentence "5 less" into an equation we can simplify, where the two opposite changes (+1 and -1) collapse neatly. Once we solve for s, tool #3 (Eliminate Possibilities) matches the cube's volume to exactly one of the five answer choices.

1STEP 1

Name the cube's edge

Let s be one edge of the cube. Because it is a cube, all three edges equal s.

s = edge length of the cube
2STEP 2

Write both volumes

The cube's volume is s³. The new box is s+1 by s-1 by s, so its volume is (s+1)(s-1)s.

V_cube = s³, V_box = (s+1)(s-1) s
3STEP 3

Simplify the box's volume

Since +1 and -1 are opposites, (s+1)(s-1) = s² - 1, so the box holds s(s² - 1) = s³ - s.

(s+1)(s-1) = s² - 1, V_box = s(s² - 1) = s³ - s
4STEP 4

Set up and solve for the edge

The drop is 5, so s³ - (s³ - s) = 5. The s³ terms cancel, leaving s = 5.

s³ - (s³ - s) = 5 → s = 5
5STEP 5

Find the cube's volume

The edge is 5, so the cube's volume is 5³ = 125, which is choice (D).

V_cube = 5³ = 125 → (D)
Answer
125
Check the numbers directly: an edge of 5 makes a cube of volume 5³ = 125, and the new box measures 6 × 4 × 5 = 120. The difference is 125 - 120 = 5, exactly as stated. The volume 125 is answer choice (D), so the result is consistent with both the problem and the choices.
💡Key takeaway

When you stretch one side by 1 and shrink another by 1, the volume drops by exactly the length of the side you left alone.

  • Name the cube's edge
  • Write both volumes
  • Simplify the box's volume
  • Set up and solve for the edge
  • Find the cube's volume

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