AMC 8 · 2019 · #9

Grade 8 geometry-3drate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

volume-cylinderratio-proportionformula-substitution identify-subproblemsformula-substitution ↑ Prerequisites: area-circlesratio-proportion

📏 Short solution 💡 2 insights

Problem

Alex's cat food can is a cylinder with diameter 6 cm and height 12 cm (tall and skinny). Felicia's can is a cylinder with diameter 12 cm and height 6 cm (short and wide). Find the ratio of Alex's volume to Felicia's volume and match it to one of the five answer choices.

Pick an answer.

(A)

1:4

(B)

1:2

(C)

1:1

(D)

2:1

(E)

4:1

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question "ratio of two volumes" naturally splits into three subproblems (Tool #7): (i) compute V_A, (ii) compute V_F, (iii) form and simplify the ratio. Tool #8 (Analyze the Units) keeps us honest about diameter-vs-radius (the formula wants r, not d) and reminds us that both volumes are in cm³, so π and cm³ cancel in the ratio. Tool #3 (Eliminate Possibilities) is the multiple-choice safety net: once we see Felicia's can is wider where it counts (radius is squared) but only half as tall, we expect V_F ＞ V_A, which already eliminates (C), (D), (E).

1STEP 1

The formula V = π r² h needs the radius, so halve each diameter: r_A = 3 cm, r_F = 6 cm.

r_A =

\frac{6}{2}

= 3 cm, r_F =

\frac{12}{2}

= 6 cm

2STEP 2

Square Alex's radius and multiply by his height: V_A = π·9·12 = 108π cm³.

V_A = π (3)² (12) = π · 9 · 12 = 108π cm³

3STEP 3

Do the same for Felicia (r_F = 6, h_F = 6): V_F = π·36·6 = 216π cm³.

V_F = π (6)² (6) = π · 36 · 6 = 216π cm³

4STEP 4

Divide the volumes; π and cm³ cancel, leaving $\frac{V_A}{V_F}$ = $\frac{108}{216}$ .

\frac{V_A}{V_F}

\frac{108π}{216π}

\frac{108}{216}

5STEP 5

Since 216 = 2 × 108, the fraction reduces to $\frac{1}{2}$ , so the ratio is 1 : 2 — choice (B).

\frac{108}{216}

\frac{1}{2}

→ V_A : V_F = 1 : 2 → (B)

Answer

1:2

Sanity check by scaling: Felicia's radius is doubled compared to Alex's (6 vs 3), so r² becomes 4× larger, but her height is halved (6 vs 12), giving an overall factor of 4 ×

\frac{1}{2}

= 2 in Felicia's favor. So V_F = 2 V_A, meaning V_A : V_F = 1:2 — exactly choice (B). The squared-radius effect outweighing the halved height matches intuition: short-and-wide cans usually hold more than tall-and-skinny ones with these flipped dimensions.

💡Key takeaway

This AMC 8 problem really only needs the Grade 8 cylinder volume formula V = π r² h — once you plug in, π cancels and the ratio simplifies in one step!

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