The curved surface area $A$ of a right circular cylinder is given by the formula:

$A = 2 \pi r h$

where:

• $r$ is the base radius of the cylinder, and
• $h$ is the height of the cylinder.

Step 1: Initial curved surface area

Let the initial radius and height of the cylinder be $r_1$ and $h_1$, respectively. So the initial curved surface area is:

$A_{\text{initial}} = 2 \pi r_1 h_1$

Step 2: Modified dimensions

The problem states that the radius is decreased by 20%. So the new radius, $r_2$, is:

$r_2 = 0.8 r_1$

The height is increased by $r\%$, so the new height, $h_2$, is:

$h_2 = h_1 \left( 1 + \frac{r}{100} \right)$

Step 3: Final curved surface area

After the changes, the new curved surface area is:

$A_{\text{final}} = 2 \pi r_2 h_2 = 2 \pi (0.8 r_1) \left( h_1 \left( 1 + \frac{r}{100} \right) \right)$

Simplifying this expression:

$A_{\text{final}} = 2 \pi (0.8 r_1) \left( h_1 \left( 1 + \frac{r}{100} \right) \right) = 2 \pi r_1 h_1 \cdot 0.8 \left( 1 + \frac{r}{100} \right)$

Step 4: Setting initial and final areas equal

Since the curved surface area remains unchanged, we equate the initial and final areas:

$2 \pi r_1 h_1 = 2 \pi r_1 h_1 \cdot 0.8 \left( 1 + \frac{r}{100} \right)$

Dividing both sides by $2 \pi r_1 h_1$:

$1 = 0.8 \left( 1 + \frac{r}{100} \right)$

Step 5: Solving for $r$

Now solve for $r$:

$1 = 0.8 + 0.8 \cdot \frac{r}{100}$

$1 - 0.8 = 0.8 \cdot \frac{r}{100}$

$0.2 = 0.8 \cdot \frac{r}{100}$

$\frac{r}{100} = \frac{0.2}{0.8} = 0.25$

$r = 25$

Final Answer:

The value of $r$ is $\boxed{25}$.

Would you like a detailed explanation of any step?

Related Questions:

1. What is the formula for the total surface area of a cylinder?
2. How does changing the radius affect the volume of a cylinder?
3. If the height is doubled and the radius is halved, how does the volume change?
4. Can the surface area and volume both remain constant if both radius and height are changed?
5. How is the concept of percentage change used in geometry problems?

Tip:

Always ensure that you properly account for percentage changes by converting them into decimal multipliers before applying them in formulas.