AMC 10 · 2015 · #7
Grade 8 geometry-3dPick an answer.
The words give a relationship, not numbers, so Tool #4 (Introduce a Variable) names the two radii and two heights and writes each volume with the same formula. Equal volume then becomes one equation. Tool #13 (Convert to Algebra) does the work inside that equation: the shared π r₁² cancels and 1.1² collapses to a single factor, leaving the two heights compared directly. Finally Tool #3 (Eliminate Possibilities) reads that factor as a percent and matches it to exactly one of the five choices while exposing the 10% trap.
Name the parts and set volumes equal
Equal volumes give one equation.
Writing both volumes with the same formula lets one equation hold both cylinders at once.
8.G.C.9Introduce A VariableTurn 10% more into a multiplier
The percentage becomes a plain multiplier.
A percent increase is just a fixed multiplier, so "10% more" is exactly × 1.1.
A percent increase is a fixed multiplier, so ten percent more is exactly one and a tenth times as much.
▸ Why?
A percent is a count out of a hundred, so the original plus that many hundredths is one fixed factor.
▸ Why?
That factor lands on the radius, and a base area picks up the radius twice over.
Cancel the common factors
Cancelling leaves a factor of 1.21.
Whatever is identical on both sides drops out, leaving the two heights compared directly.
7.EE.B.4Convert To AlgebraRead 1.21 as a percent and pick the choice
Read as a percent, that is 21 percent more, choice (D).
A factor of 1.21 is the same as adding 21% on top of the second height.
7.RP.A.3Eliminate PossibilitiesWhen the volume stays fixed, height has to fight the radius squared, so a 10% wider radius opens a 21% gap in height, not a 10% one.
- Name the parts and set volumes equal
- Turn 10% more into a multiplier
- Cancel the common factors
- Read 1.21 as a percent and pick the choice