* The volume of a cylinder is directly proportional to its height and to the square of its radius.
If the height is halved while the radius is tripled, what will be the percentage change in the volume of the cylinder?

The volume \( V \) of a cylinder is given by the formula:

\[
V = \pi r^2 h
\]

where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder
- \( \pi \) is a constant (approximately 3.14159)

Now, we are given that the height is halved and the radius is tripled. Let's analyze the effect of these changes on the volume.

### Original volume:
\[
V_{\text{original}} = \pi r^2 h
\]

### New dimensions:
- The new height is \( \frac{h}{2} \).
- The new radius is \( 3r \).

### New volume:
\[
V_{\text{new}} = \pi (3r)^2 \left( \frac{h}{2} \right)
\]
Simplifying the expression:
\[
V_{\text{new}} = \pi (9r^2) \left( \frac{h}{2} \right)
\]
\[
V_{\text{new}} = \frac{9}{2} \pi r^2 h
\]

### Change in volume:
To find the factor by which the volume changes, we compare the new volume to the original volume:
\[
\frac{V_{\text{new}}}{V_{\text{original}}} = \frac{\frac{9}{2} \pi r^2 h}{\pi r^2 h} = \frac{9}{2}
\]
So the volume becomes \( \frac{9}{2} \) or 4.5 times the original volume.

### Percentage change:
The percentage change in volume is calculated as:
\[
\text{Percentage change} = \left( \frac{V_{\text{new}} - V_{\text{original}}}{V_{\text{original}}} \right) \times 100 = \left( \frac{\frac{9}{2} - 1}{1} \right) \times 100 = \left( 4.5 - 1 \right) \times 100 = 3.5 \times 100 = 350\%
\]

Thus, the volume of the cylinder increases by **350%**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How does the volume of a cone compare to that of a cylinder with the same base and height?
2. If both height and radius of a cylinder are doubled, how does the volume change?
3. What is the relationship between surface area and volume in a cylinder?
4. How would the volume of a cylinder change if the radius is doubled but the height remains constant?
5. Can you find a real-life example where the cylinder volume formula is used?

**Tip:** When dimensions of a geometric shape change, calculating volume ratios can help identify the percentage change efficiently.

The volume of a cylinder is directly proportional to its height and to the square of its radius. If the height is halved while the radius is tripled, what will be the percentage change in the volume of the cylinder?

Learn how to calculate the percentage change in the volume of a cylinder when its height is halved and the radius is tripled. This problem involves geometry, volume formulas, and proportionality concepts, suitable for students in Grades 8-10.

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