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To solve part (a), we need to determine the proportion of the volume of the cylinder that is not filled by the spheres.

### Step 1: Volume of the Cylinder
The volume \( V_{\text{cylinder}} \) of a cylinder is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder.

In the problem, the height of the cylinder is given as \( h = 6r \), so:
\[
V_{\text{cylinder}} = \pi r^2 (6r) = 6\pi r^3
\]

### Step 2: Volume of the 3 Spheres
The volume \( V_{\text{sphere}} \) of a single sphere is given by the formula:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Since there are 3 spheres, the total volume of the spheres is:
\[
V_{\text{spheres}} = 3 \times \frac{4}{3} \pi r^3 = 4 \pi r^3
\]

### Step 3: Proportion of the Cylinder Not Filled by Spheres
To find the volume of the cylinder that is not filled by the spheres, subtract the volume of the spheres from the volume of the cylinder:
\[
V_{\text{empty}} = V_{\text{cylinder}} - V_{\text{spheres}} = 6\pi r^3 - 4\pi r^3 = 2\pi r^3
\]

The proportion of the cylinder that is not filled by the spheres is the ratio of the empty volume to the total volume of the cylinder:
\[
\text{Proportion not filled} = \frac{V_{\text{empty}}}{V_{\text{cylinder}}} = \frac{2\pi r^3}{6\pi r^3} = \frac{2}{6} = \frac{1}{3}
\]

Thus, the proportion of the cylinder that is not filled by the spheres is **\(\frac{1}{3}\)**.

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Would you like any additional details or clarifications on the solution? 

Here are five related questions to deepen your understanding:
1. How would the proportion change if the number of spheres increased to 4?
2. What is the total surface area of the spheres inside the cylinder?
3. How does the volume formula for a sphere relate to that of a cylinder?
4. What happens to the volume proportion if the height of the cylinder is \(7r\)?
5. Can we fit any larger spheres in the same cylinder without changing the radius?

**Tip:** Always double-check the dimensions and formulas you are using to ensure they align with the problem's conditions.

A hollow cylinder has radius r cm and height 6r cm. 3 spheres, also of radius r cm, are put into the cylinder. Work out the proportion of the cylinder that is not filled by the spheres.

Learn how to calculate the proportion of a hollow cylinder that is not filled by 3 spheres of equal radius. This geometry problem uses key volume formulas for cylinders and spheres, suitable for Grades 10-12.

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