A spherical container with a diameter of 30 cm is 3/4 full of water. If the water is emptied into a cylindrical container with a diameter of 12 cm, what is the depth of the water in the cylindrical co

The diameter of the spherical container is 30 cm, so its radius $r_s$ is $\frac{30}{2} = 15$ cm.
The formula for the volume of a sphere is $V_s = \frac{4}{3}\pi r_s^3$.

$$V_s = \frac{4}{3}\pi (15 \text{ cm})^3$$ $$V_s = \frac{4}{3}\pi (3375 \text{ cm}^3)$$ $$V_s = 4 \pi (1125 \text{ cm}^3)$$ $$V_s = 4500\pi \text{ cm}^3$$

The spherical container is $\frac{3}{4}$ full of water.
The volume of water $V_w$ is $\frac{3}{4}$ of the total volume of the sphere.

$$V_w = \frac{3}{4} \times 4500\pi \text{ cm}^3$$ $$V_w = 3 \times 1125\pi \text{ cm}^3$$ $$V_w = 3375\pi \text{ cm}^3$$

The water is emptied into a cylindrical container with a diameter of 12 cm.
The radius of the cylindrical container $r_c$ is $\frac{12}{2} = 6$ cm.
The formula for the volume of a cylinder is $V_c = \pi r_c^2 h$, where $h$ is the depth of the water.
Since the volume of water remains the same, $V_w = V_c$.

$$3375\pi \text{ cm}^3 = \pi (6 \text{ cm})^2 h$$ $$3375\pi \text{ cm}^3 = 36\pi \text{ cm}^2 h$$

Divide both sides by $36\pi \text{ cm}^2$ to solve for $h$:

$$h = \frac{3375\pi \text{ cm}^3}{36\pi \text{ cm}^2}$$ $$h = \frac{3375}{36} \text{ cm}$$ $$h = 93.75 \text{ cm}$$

The depth of the water in the cylindrical container is $93.75 \text{ cm}$.