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To solve this problem, we'll break it down into steps:

### Given:
- Four spheres of radius \( r = 12 \) cm are inside a cylinder.
- The height of the cylinder is the sum of the diameters of the four spheres, since they just fit inside.
- The diameter of each sphere \( = 2r = 24 \) cm.
- The total height of the cylinder \( = 4 \times 24 = 96 \) cm.
- The radius of the cylinder is equal to the radius of the spheres, so the radius \( R \) of the cylinder is \( 12 \) cm.

### Step 1: Volume of the Cylinder
The volume of the cylinder \( V_{\text{cyl}} \) is given by the formula:
\[
V_{\text{cyl}} = \pi R^2 h
\]
Substituting the values:
\[
V_{\text{cyl}} = \pi (12)^2 \times 96 = 13824\pi \text{ cubic centimeters}
\]

### Step 2: Volume of the Spheres
The volume of one sphere \( V_{\text{sphere}} \) is given by:
\[
V_{\text{sphere}} = \frac{4}{3}\pi r^3
\]
Substituting the value of \( r \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi \times 1728 = 2304\pi \text{ cubic centimeters}
\]
Since there are four spheres:
\[
\text{Total volume of the spheres} = 4 \times 2304\pi = 9216\pi \text{ cubic centimeters}
\]

### Step 3: Volume of Empty Space in the Cylinder
The volume of the empty space inside the cylinder is the difference between the volume of the cylinder and the total volume of the spheres:
\[
V_{\text{empty}} = V_{\text{cyl}} - \text{Total volume of the spheres}
\]
Substituting the calculated values:
\[
V_{\text{empty}} = 13824\pi - 9216\pi = 4608\pi \text{ cubic centimeters}
\]

So, the volume of the empty space inside the cylinder is \( 4608\pi \) cubic centimeters.

Would you like more details or have any questions? Here are some related questions:

1. How would the solution change if the radius of the spheres were different?
2. What if there were more or fewer spheres fitting inside the cylinder?
3. How do you derive the volume formula for a cylinder?
4. Could this problem be solved using integration? If so, how?
5. What would happen if the cylinder's radius was larger than the spheres' radius?

**Tip:** When working with volumes, remember that the volume of composite shapes can often be found by adding or subtracting the volumes of simpler shapes.

Four spheres of radius 12 cm just fit inside a cylinder. Calculate the volume inside the cylinder that is empty. Leave your answer in terms of π.

Learn how to calculate the empty volume inside a cylinder that contains four spheres, each with a radius of 12 cm. This problem involves applying geometric formulas for the volume of a cylinder and spheres. Suitable for students in Grades 9-12.

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