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.