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Let's analyze and solve the problem.

### Problem Statement:
- A cone is fixed on top of a cylinder.
- The radius of both the cone and the cylinder is \( r \) cm.
- The height of the cone is \( 5r \) cm.
- The total height of the decoration (cone + cylinder) is \( 6r \) cm.
- The total volume of the decoration is \( 225 \) cm³.

### Solution Steps:
1. **Determine the height of the cylinder:**
   - Since the total height is \( 6r \) cm and the cone’s height is \( 5r \) cm, the cylinder’s height is:
     \[
     h_{\text{cylinder}} = 6r - 5r = r \text{ cm}
     \]

2. **Volume of the cone:**
   - The volume \( V \) of a cone is given by:
     \[
     V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (5r)
     \]
     Simplifying this:
     \[
     V_{\text{cone}} = \frac{5}{3} \pi r^3
     \]

3. **Volume of the cylinder:**
   - The volume \( V \) of a cylinder is given by:
     \[
     V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (r) = \pi r^3
     \]

4. **Total volume of the decoration:**
   - Adding the volumes of the cone and cylinder:
     \[
     V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = \frac{5}{3} \pi r^3 + \pi r^3
     \]
     Simplifying this:
     \[
     V_{\text{total}} = \left(\frac{5}{3} + 1\right) \pi r^3 = \frac{8}{3} \pi r^3
     \]

5. **Set the total volume equal to the given volume:**
   \[
   \frac{8}{3} \pi r^3 = 225
   \]

6. **Solve for \( r \):**
   \[
   r^3 = \frac{225 \times 3}{8 \pi}
   \]
   \[
   r^3 = \frac{675}{8 \pi}
   \]
   Calculating the right-hand side:
   \[
   r = \sqrt[3]{\frac{675}{8 \pi}}
   \]

Let's calculate \( r \).The value of \( r \) is approximately \( 3.00 \) cm when rounded to three significant figures.

### Summary:
- The radius \( r \) of the base of the cone and cylinder is \( 3.00 \) cm.

Would you like more details or have any questions about this solution?

### Further Questions:
1. How do the volumes of a cone and cylinder compare when their heights and radii are related?
2. What happens to the volume of the decoration if the height of the cone changes?
3. How would the volume formula change if the cylinder's height was different?
4. Can you solve a similar problem with a different total volume?
5. How does changing the radius \( r \) affect the overall height of the decoration?

### Tip:
Always check units carefully when dealing with geometric shapes; consistency is key to avoiding calculation errors.

The base of a cone is fixed to the top of a cylinder to make a decoration. The radius of the base of the cone and of the cylinder is r cm. The cone's height is 5r cm. The total height of the decoration is 6r cm. The total volume of the decoration is 225 cm³. Calculate the value of r, giving your answer correct to 3 significant figures.

Learn how to calculate the radius of a decoration made by fixing a cone on top of a cylinder, given specific heights and a total volume. This problem involves geometry and algebra, including the application of volume formulas for a cone and a cylinder. Suitable for students in Grades 10-12.

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