Answer
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.
Solution
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.
Reviewed and approved by the UpStudy tutoring team
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