What is the relationship between the radius and height of a cylinder when calculating its volume?

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The volume $$ V $$ of a cylinder is determined by both its **radius** $$ r $$ and its **height** $$ h $$ using the following formula:

$$
V = \pi r^2 h
$$

### Breakdown of the Relationship:

1. **Radius ($$ r $$)**:
   - The radius appears squared ($$ r^2 $$) in the formula, indicating that the volume increases **quadratically** as the radius increases.
   - **Implication**: Doubling the radius results in the volume increasing by a factor of four ($$ 2^2 = 4 $$).

2. **Height ($$ h $$)**:
   - The height is a linear factor in the formula.
   - **Implication**: Doubling the height results in the volume doubling as well.

### Direct Proportionality:

- **Volume and Radius**: $$ V \propto r^2 $$
- **Volume and Height**: $$ V \propto h $$

This means:
- If you **increase the radius**, the volume increases by the square of the change in radius, assuming the height remains constant.
- If you **increase the height**, the volume increases linearly with the height, assuming the radius remains constant.

### Practical Example:

- **Given**: Radius $$ r = 3 $$ units, Height $$ h = 5 $$ units.
- **Volume**: $$ V = \pi \times 3^2 \times 5 = 45\pi $$ cubic units.

### Visualization:

Imagine two cylinders:
- **Cylinder A**: Radius $$ r $$, Height $$ h $$
- **Cylinder B**: Radius $$ 2r $$, Height $$ h $$

**Volume of Cylinder B**:
$$
V_B = \pi (2r)^2 h = \pi \times 4r^2 \times h = 4V_A
$$
Cylinder B has four times the volume of Cylinder A because its radius is doubled.

### Conclusion:

When calculating the volume of a cylinder:
- **Radius and height are both crucial**, with the radius affecting the volume exponentially (quadratically) and the height affecting it linearly.
- Understanding their relationship helps in optimizing or scaling cylindrical shapes for various applications, such as storage tanks, pipes, and more.

**Formula Recap**:
$$
V = \pi r^2 h
$$

This formula succinctly captures how the radius and height of a cylinder contribute to its overall volume.
The volume of a cylinder depends on both its radius and height. The formula is $$ V = \pi r^2 h $$. This means the volume increases quadratically with the radius and linearly with the height.

What is the relationship between the radius and height of a cylinder when calculating its volume?

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