AMC 10 · 2009 · #5
Grade 7 geometry-3dPick an answer.
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.
Name the cube's edge
Naming the edge hangs everything on one letter.
Giving the unknown edge a name lets you describe both solids with a single symbol.
6.EE.B.6Introduce A VariableWrite both volumes
Both volumes can then be written down.
Volume is just the three side lengths multiplied together.
5.MD.C.5Convert To AlgebraSimplify the box's volume
The changed edges multiply into a difference of squares.
The two opposite tweaks, +1 and -1, cancel into a clean s² - 1.
The two opposite tweaks, one up and one down, cancel into a clean square minus one.
▸ Why?
A number one bigger times a number one smaller is a difference of two squares, so it collapses at once.
▸ Why?
Multiplying the brackets out spreads each term across the other, which is what makes the middle terms cancel.
Set up and solve for the edge
Subtracting leaves a one-step equation.
Because the cube parts cancel, the whole 5-unit drop is exactly one edge length.
6.EE.B.7Introduce A VariableFind the cube's volume
Cubing the edge gives 125, choice (D).
Cubing the edge you found gives the volume the question asked for.
6.EE.A.1Eliminate PossibilitiesWhen 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