If the radius of a cylinder is tripled but its curved surface area is unchanged, then its height will be

Options

A. tripled
B. constant
C. one sixth
D. one third

Answer

To solve the problem, we need to find the height of a cylinder when its radius is tripled while keeping its curved surface area unchanged.

### Step-by-Step Solution:

1. **Understand the formula for the curved surface area of a cylinder**:
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

2. **Set up the initial curved surface area**:
Let the original radius be \( r \) and the original height be \( h \). Therefore, the original curved surface area is:
\[ \text{CSA} = 2 \pi r h \]

3. **Consider the new radius**:
If the radius is tripled, the new radius becomes:
\[ r' = 3r \]

4. **Set up the equation for the new curved surface area**:
Since the curved surface area remains unchanged, we can write the equation for the new curved surface area with the new radius and the new height \( H \):
\[ \text{CSA} = 2 \pi (3r) H \]
This simplifies to:
\[ \text{CSA} = 6 \pi r H \]

5. **Equate the two expressions for curved surface area**:
Since the curved surface area is unchanged, we can set the two expressions equal to each other:
\[ 2 \pi r h = 6 \pi r H \]

6. **Cancel out common terms**:
We can divide both sides of the equation by \( 2 \pi r \) (assuming \( r \neq 0 \)):
\[ h = 3H \]

7. **Solve for the new height \( H \)**:
Rearranging the equation gives:
\[ H = \frac{h}{3} \]

### Conclusion:
Thus, the new height \( H \) of the cylinder when the radius is tripled and the curved surface area remains unchanged is:
\[ H = \frac{h}{3} \]