2021 AMC 10B Problem 10

Pouring water from one container to another does not change how much water there is. So the volume of the cone equals the volume of the water in the cylinder.

1. Volume of the cone. A cone's volume is one third of base area times height:
$$\tfrac{1}{3} \times \pi \times 12^2 \times 18 = \tfrac13 \times 144 \times 18 \times \pi = 864\pi.$$
2. Volume of the water in the cylinder. It is a short cylinder with radius $24$ and unknown height $h$:
$$\pi \times 24^2 \times h = 576\pi h.$$
3. Set them equal. $576\pi h = 864\pi$, so
$$h = \frac{864}{576} = \frac{3}{2} = 1.5.$$

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

"Poured into" always means "same volume." You can also see the answer with ratios: the cylinder's radius is $2$ times the cone's, so its base is $4$ times as big, and the cone only holds one third of a cylinder with its own base. So the water height is $18 \div 4 \div 3 = 1.5$.

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

In the cylinder this water forms a cylinder of radius $24$ and unknown height $h$:
$$\pi (24)^2 h = 576\pi h = 864\pi \quad\Longrightarrow\quad h = \frac{864}{576} = \frac{3}{2}.$$

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