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.

Let's analyze the changes step by step:

1. **Original Volume:**
   \[
   V_{\text{original}} = \pi r^2 h
   \]

2. **New Volume after changes:**
   - The radius \( r \) is tripled, so the new radius is \( 3r \).
   - The height \( h \) is halved, so the new height is \( \frac{h}{2} \).

   Substituting these into the volume formula gives the new volume \( V_{\text{new}} \):

   \[
   V_{\text{new}} = \pi (3r)^2 \left(\frac{h}{2}\right)
   \]

   Simplifying this expression:

   \[
   V_{\text{new}} = \pi \cdot 9r^2 \cdot \frac{h}{2} = \frac{9}{2} \cdot \pi r^2 h
   \]

3. **Comparing the new volume to the original volume:**

   \[
   \frac{V_{\text{new}}}{V_{\text{original}}} = \frac{\frac{9}{2} \cdot \pi r^2 h}{\pi r^2 h} = \frac{9}{2} = 4.5
   \]

   This means the new volume is 4.5 times the original volume.

4. **Calculating the percentage change:**

   The percentage change in volume is given by:

   \[
   \text{Percentage Change} = \left(\frac{V_{\text{new}} - V_{\text{original}}}{V_{\text{original}}}\right) \times 100\%
   \]

   Substituting the values:

   \[
   \text{Percentage Change} = \left(\frac{4.5V_{\text{original}} - V_{\text{original}}}{V_{\text{original}}}\right) \times 100\% = \left(4.5 - 1\right) \times 100\% = 3.5 \times 100\% = 350\%
   \]

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

Would you like further details or have any other questions?

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Here are 5 related questions you might find interesting:
1. How does the volume change if both the radius and height are doubled?
2. What happens to the volume if the radius is halved and the height remains the same?
3. How would the surface area of the cylinder change if the height is halved and the radius is tripled?
4. If the radius is kept constant but the height is tripled, by what percentage does the volume change?
5. How do changes in radius and height affect the lateral surface area of the cylinder?

**Tip:** Always remember that in problems involving direct proportionality, changes in one dimension have a multiplicative effect on the volume, especially when they involve squares or cubes, as seen in cylinder volume calculations.

Understand how changes in cylinder dimensions affect its volume. Learn to calculate percentage changes when height is halved and radius is tripled using the formula for cylinder volume. Suitable for high school students.

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