AMC 10 · 2003 · #13

Grade 8 geometry-3d

Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid

volume-spherevolume-conepercentage convert-to-algebraformula-substitution ↑ Prerequisites: volume-sphere

📏 Long solution 💡 2 insights

Problem

A ball of ice cream sits on a right circular cone whose base circle has the same diameter as the ball. When the ice cream melts it shrinks to 75% of its frozen volume, and that melted amount exactly fills the cone. Find the ratio of the cone's height to its radius.

Pick an answer.

(A)

2:1

(B)

3:1

(C)

4:1

(D)

16:3

(E)

6:1

How to solve

Strategy Introduce a Variable

No number is given anywhere, so the first move is to name the two lengths that matter: the shared radius r and the cone height h (Tool #4). The clause "same diameter as the sphere" is what collapses three possible lengths into two, and picturing the ball resting on the cone makes that sharing obvious (Tool #10). "Exactly fills" is a volume equation in disguise, so write both volumes with the standard formulas and set them equal (Tool #13). Because the question asks only for a ratio, π and r will divide out, and anyone who wants a concrete check can set r=1 and compare two plain numbers (Tool #9).

1STEP 1

Name the two lengths

Equal diameters give equal radii, so one letter r serves both shapes.

r=sphere radius=cone base radius, h=cone height

2STEP 2

Write the frozen volume

The frozen scoop is a sphere of volume four thirds pi r cubed.

V_sphere=4/3π r³

3STEP 3

Take 75 percent — the fractions cancel

The shrink factor and that fraction are reciprocals, so the melt is exactly pi r cubed.

V_melted=3/4·4/3π r³=π r³

4STEP 4

Set the cone equal to the puddle

Filling exactly means the cone's volume equals that, giving one equation.

1/3π r²h=π r³

5STEP 5

Solve for the height

Dividing out the common factors leaves h = 3r.

1/3π r²h=π r³ → 1/3h=r → h=3r

6STEP 6

Read off the ratio

So the ratio is 3:1, independent of size, choice (B).

h:r=3r:r=3:1; r=1: 3/4·4/3π=π=1/3π(1)²(3)

Answer

3:1

The numbers check exactly, not approximately: with r=1 the melted volume is π and the cone of height 3 holds 1/3π · 1 · 3=π. The choice list also explains itself. If the melting step were skipped and the full sphere had to fit, the equation would be 1/3π r²h=4/3π r³, giving h=4r and choice (C) 4:1 — so (C) is exactly the trap for forgetting the 75%, and the true answer must be smaller than 4:1, which 3:1 is. Choice (D) 16:3 and (E) 6:1 are both larger than 4:1, meaning they would need more melted ice cream than there was frozen ice cream, which is impossible when melting shrinks the volume. That leaves only (A) and (B) as physically sensible, and the algebra picks (B).

💡Key takeaway

Three quarters of a sphere's volume is exactly π r³, and a cone of the same radius has to stand 3 radii tall to hold that much.

Name the two lengths
Write the frozen volume
Take 75 percent — the fractions cancel
Set the cone equal to the puddle
Solve for the height
Read off the ratio