Increasing the radius of a cylinder by 6 units increases the volume by y cubic units. Increasing the altitude of the cylinders by 6 units also increases the volume by y cubic units. If the original altitude is 2, then the original radius is ?

Options

A. 8 units
B. 4 units
C. 6 units
D. 5 units

Answer

To solve the problem, we need to find the original radius of a cylinder given that increasing the radius and the altitude both lead to the same increase in volume. Let's denote the original radius as \( r \) and the original altitude (height) as \( h = 2 \).

### Step-by-Step Solution:

1. **Volume of the Cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting \( h = 2 \):
\[ V = \pi r^2 \cdot 2 = 2\pi r^2 \]

2. **Increasing the Radius**:
When the radius is increased by 6 units, the new radius becomes \( r + 6 \). The new volume \( V' \) is:
\[ V' = \pi (r + 6)^2 \cdot 2 \]
Expanding this:
\[ V' = 2\pi (r^2 + 12r + 36) = 2\pi r^2 + 24\pi r + 72\pi \]
The increase in volume when the radius is increased is:
\[ \Delta V_r = V' - V = (2\pi r^2 + 24\pi r + 72\pi) - 2\pi r^2 = 24\pi r + 72\pi \]
We denote this increase as \( y \):
\[ y = 24\pi r + 72\pi \]

3. **Increasing the Altitude**:
When the altitude is increased by 6 units, the new altitude becomes \( h + 6 = 2 + 6 = 8 \). The new volume \( V'' \) is:
\[ V'' = \pi r^2 \cdot 8 = 8\pi r^2 \]
The increase in volume when the altitude is increased is:
\[ \Delta V_h = V'' - V = 8\pi r^2 - 2\pi r^2 = 6\pi r^2 \]
We denote this increase as \( y \):
\[ y = 6\pi r^2 \]

4. **Setting the Two Expressions for \( y \) Equal**:
Since both expressions for \( y \) are equal, we can set them equal to each other:
\[ 24\pi r + 72\pi = 6\pi r^2 \]

5. **Simplifying the Equation**:
Dividing the entire equation by \( 6\pi \) (since \( \pi \) is non-zero):
\[ 4r + 12 = r^2 \]
Rearranging gives:
\[ r^2 - 4r - 12 = 0 \]

6. **Factoring the Quadratic Equation**:
To factor the quadratic equation \( r^2 - 4r - 12 = 0 \), we look for two numbers that multiply to \(-12\) and add to \(-4\). The factors are \(-6\) and \(2\):
\[ (r - 6)(r + 2) = 0 \]

7. **Finding the Values of \( r \)**:
Setting each factor to zero gives:
\[ r - 6 = 0 \quad \Rightarrow \quad r = 6 \]
\[ r + 2 = 0 \quad \Rightarrow \quad r = -2 \quad (\text{not valid since radius cannot be negative}) \]

8. **Conclusion**:
The original radius of the cylinder is:
\[ r = 6 \text{ units} \]