Find the maximum volume of right circular cylinder, if the sum of its radius and height is 6 units.

Options

A. `24 pi`
B. `28 pi`
C. `32 pi`
D. None of These

Explanation

To find the maximum volume of a right circular cylinder given that the sum of its radius (r) and height (h) is 6 units, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to maximize the volume \( V \) of a right circular cylinder, which is given by the formula:
\[ V = \pi r^2 h \]
We also know that the sum of the radius and height is:
\[ r + h = 6 \]

2. **Express Height in Terms of Radius**: From the equation \( r + h = 6 \), we can express height \( h \) in terms of radius \( r \):
\[ h = 6 - r \]

3. **Substitute for Height in Volume Formula**: Now substitute \( h \) in the volume formula:
\[ V = \pi r^2 (6 - r) \]
Simplifying this, we get:
\[ V = \pi r^2 (6 - r) = 6\pi r^2 - \pi r^3 \]

4. **Differentiate the Volume Function**: To find the maximum volume, we need to take the derivative of \( V \) with respect to \( r \) and set it to zero:
\[ \frac{dV}{dr} = \frac{d}{dr}(6\pi r^2 - \pi r^3) \]
Using the power rule:
\[ \frac{dV}{dr} = 12\pi r - 3\pi r^2 \]

5. **Set the Derivative to Zero**: Set the derivative equal to zero to find critical points:
\[ 12\pi r - 3\pi r^2 = 0 \]
Factoring out \( 3\pi r \):
\[ 3\pi r(4 - r) = 0 \]

6. **Solve for Radius**: This gives us two solutions:
\[ 3\pi r = 0 \quad \text{or} \quad 4 - r = 0 \]
Thus, \( r = 0 \) or \( r = 4 \).

7. **Determine Maximum Volume**: We need to check which of these values gives a maximum volume:
- If \( r = 0 \), then \( h = 6 \) and \( V = 0 \).
- If \( r = 4 \), then \( h = 6 - 4 = 2 \) and:
\[ V = \pi (4^2)(2) = \pi (16)(2) = 32\pi \]

8. **Conclusion**: The maximum volume of the cylinder is:
\[ V = 32\pi \text{ cubic units} \]