A cylindrical container with an internal radius of 12 cm is filled with water to a height H. A sphere with a radius of 12 cm is placed at the bottom of this container, and we find that the water covers exactly the level of the ball. How much is the height H in cm?

To solve the problem, we need to determine the height \( H \) of the water when a sphere with a radius of 12 cm is fully submerged in a cylindrical container with the same internal radius. The total volume of water is made up of the volume of the submerged sphere plus the initial volume of the water.

### Step 1: Volume of the Sphere
The volume \( V_{\text{sphere}} \) of a sphere is given by the formula:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Given that the radius \( r = 12 \) cm, the volume of the sphere is:
\[
V_{\text{sphere}} = \frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi (1728) = 2304\pi \text{ cubic centimeters}
\]

### Step 2: Volume of the Water in the Cylinder
The volume \( V_{\text{cylinder}} \) of the water in the cylinder up to height \( H \) is given by:
\[
V_{\text{cylinder}} = \pi r^2 H
\]
With the radius \( r = 12 \) cm, this becomes:
\[
V_{\text{cylinder}} = \pi (12)^2 H = 144\pi H \text{ cubic centimeters}
\]

### Step 3: Setting up the Equation
Since the water level just covers the sphere, the volume of water must equal the volume of the submerged sphere:
\[
144\pi H = 2304\pi
\]

### Step 4: Solve for \( H \)
We can cancel out \( \pi \) from both sides:
\[
144H = 2304
\]
Divide both sides by 144:
\[
H = \frac{2304}{144} = 16 \text{ cm}
\]

Thus, the height \( H \) of the water in the cylindrical container is **16 cm**.

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### Related Questions:
1. What would be the new height if the sphere had a radius of 10 cm?
2. How does the volume of water change if the cylinder's radius is doubled?
3. What happens to the height \( H \) if the sphere is replaced with a hemisphere?
4. How would the height change if the sphere was only half submerged?
5. What if the cylindrical container is only partially filled before adding the sphere?
6. Can you find the height \( H \) if the cylinder's radius was

Learn how to calculate the height of water in a cylindrical container containing a submerged sphere. Detailed solution provided with volume formulas for sphere and cylinder. Suitable for Grades 9-12.

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