AMC 10 · 2004 · #9

Grade 8 geometry-3d
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
volume-cylinderpercentage convert-to-algebraeasier-related-problem ↑ Prerequisites: ratio-proportion
📏 Medium solution 💡 1 insight
Problem
A peanut butter jar is a cylinder. Its diameter is made 25% larger, but the volume stays exactly the same. Find the percent by which the height must be decreased.

Pick an answer.

(A)
10
(B)
25
(C)
36
(D)
50
(E)
60
How to solve
Strategy Introduce a Variable

Nothing in the problem gives an actual radius or height, only how they change, so name the original radius r and original height h (Tool #4) and write the volume as π r² h. Then the two changes can be tracked as multipliers on those letters: the diameter growing 25% multiplies the radius, squaring turns that into the base-area multiplier, and forcing the volume to stay equal pins down the height multiplier. Because the volume formula is the same shape before and after, the π and the letters r and h cancel, leaving a clean number for the new height as a fraction of the old. Once the algebra is set up, plugging in easy concrete numbers (Tool #9) is a fast independent check that the percent is right.

1STEP 1

Name the sizes and write the volume

Name the radius and height so the volume is pi r squared h.

V = π r² h
2STEP 2

Turn the wider diameter into a radius multiplier

The radius scales with the diameter, so it is multiplied by 5/4.

r' = 1.25 r = 5/4r
3STEP 3

Square the radius to get the new base area

Area goes with the square, so the base grows by 25/16, far more than a quarter.

π (r')² = π(5/4r)² = 25/16 π r²
4STEP 4

Force equal volume and read off the height cut

Equal volume forces the height to 64% of the old, a drop of 36%, choice (C).

25/16π r² h' = π r² h → h' = 16/25h = 0.64 h → 36% decrease
Answer
36
The base got wider, so the height has to shrink — and because area depends on the square of the radius, the base grew a lot (56.25%), so the height must fall by more than the 25% the diameter rose. That rules out (A) 10 and (B) 25 as too small, while 0.64 of the old height is nowhere near a half-cut, ruling out (D) 50 and (E) 60 as too big. A 36% decrease sits exactly where it should. The trap answer is (B) 25, from wrongly assuming the height drops by the same percent the diameter rose, forgetting the squaring.
💡Key takeaway

When a circle's width grows, its area grows by the square of that factor, so to keep the same volume the height must shrink by more than the width grew.

  • Name the sizes and write the volume
  • Turn the wider diameter into a radius multiplier
  • Square the radius to get the new base area
  • Force equal volume and read off the height cut

More like this

Same archetype — closest grade level first.