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2021 AMC 10B Problem 10

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2021 AMC 10B Problem 10AMC 10 Step by Step
Name:                             Date: September 13, 2026
Solid Geometry · difficulty 1 of 5 · about 2 min · position P1-10

An inverted cone with base radius 12cm12 \text{cm} and height 18cm18\text{cm} is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of 24cm24\text{cm} . What is the height in centimeters of the water in the cylinder?

  1. A)

    1.51.5

  2. B)

    33

  3. C)

    44

  4. D)

    4.54.5

  5. E)

    66

Answer

A

Key insight

Equate volumes: (1/3)π·12²·18 = π·24²·h, so h = 864/576 = 1.5.

Solution

The volume of water equals the cone's volume:
V=13π(12)2(18)=13π⋅144⋅18=864π.V = \frac{1}{3}\pi (12)^2 (18) = \frac{1}{3}\pi \cdot 144 \cdot 18 = 864\pi.

In the cylinder this water forms a cylinder of radius 2424 and unknown height hh:
π(24)2h=576πh=864π⟹h=864576=32.\pi (24)^2 h = 576\pi h = 864\pi \quad\Longrightarrow\quad h = \frac{864}{576} = \frac{3}{2}.

The answer is (A) 1.5\boxed{\textbf{(A)}\ 1.5}.

Why this works

Pouring preserves volume, so the problem is just "cone volume equals cylinder volume." Cancel π\pi early and keep the ratios: the cylinder's radius is double the cone's, so its base area is 44 times larger, and the cone's factor 13\frac{1}{3} shrinks things further: h=183⋅4=1.5h = \frac{18}{3 \cdot 4} = 1.5.

The trap

Forgetting the 1/3 in the cone volume formula, which gives 4.5 (choice D).

Common mistakes

  • Forgetting the 1/3 in the cone volume formula, which gives 4.5 (choice D).
  • Using the ratio of radii (22) instead of the ratio of base areas (44), which gives 33.

Techniques

Set up the equation/formula and compute; no special trick needed