if the radius and height of a cylinder increase by the same percentage and the curved surface area doubles as a result of this change , find out the percentage increase in volume of cylinder

Let the initial radius and height of the cylinder be \( r \) and \( h \), respectively.

### 1. Curved Surface Area (CSA) of the Cylinder:
The curved surface area of a cylinder is given by:
\[
CSA = 2\pi r h
\]
Let both the radius and height increase by \( x\% \), meaning the new radius and height become:
\[
r' = r(1 + \frac{x}{100}) \quad \text{and} \quad h' = h(1 + \frac{x}{100})
\]
The new curved surface area becomes:
\[
CSA' = 2\pi r' h' = 2\pi r(1 + \frac{x}{100}) \cdot h(1 + \frac{x}{100}) = 2\pi r h (1 + \frac{x}{100})^2
\]
We are told that the new curved surface area is double the original:
\[
CSA' = 2 \times CSA \quad \Rightarrow \quad 2\pi r h (1 + \frac{x}{100})^2 = 2 \times 2\pi r h
\]
Simplifying:
\[
(1 + \frac{x}{100})^2 = 2
\]
Taking the square root on both sides:
\[
1 + \frac{x}{100} = \sqrt{2}
\]
\[
\frac{x}{100} = \sqrt{2} - 1
\]
\[
x = 100(\sqrt{2} - 1) \approx 100(1.414 - 1) = 41.4\%
\]
So, the radius and height increase by **41.4%**.

### 2. Volume of the Cylinder:
The volume of a cylinder is given by:
\[
V = \pi r^2 h
\]
After the increase, the new volume \( V' \) is:
\[
V' = \pi (r')^2 h' = \pi \left(r(1 + \frac{x}{100})\right)^2 \cdot h(1 + \frac{x}{100})
\]
\[
V' = \pi r^2 h (1 + \frac{x}{100})^3
\]
Substituting \( 1 + \frac{x}{100} = \sqrt{2} \):
\[
V' = \pi r^2 h (\sqrt{2})^3 = \pi r^2 h \cdot 2\sqrt{2}
\]
Thus, the new volume is \( 2\sqrt{2} \) times the original volume. Therefore, the percentage increase in volume is:
\[
(2\sqrt{2} - 1) \times 100 \approx (2.828 - 1) \times 100 = 182.8\%
\]
Thus, the volume increases by approximately **182.8%**.

Would you like more details on any part of this explanation?

Here are 5 related questions:
1. How does the volume formula of a cylinder change with respect to radius and height?
2. What is the effect of doubling the radius on the volume of a cylinder, keeping the height constant?
3. How would you solve for the new surface area if only the radius of a cylinder increases?
4. What would happen to the volume if the radius and height were tripled instead of just increased by a percentage?
5. Can you generalize this approach for a cone instead of a cylinder?

**Tip**: When dealing with percentages, always convert them into decimal form before applying them to equations to avoid calculation errors.

If the radius and height of a cylinder increase by the same percentage and the curved surface area doubles as a result of this change, find out the percentage increase in the volume of the cylinder.

Learn how to calculate the percentage increase in the volume of a cylinder when its radius and height both increase by the same percentage, leading to a doubled curved surface area. This solution involves geometry and exponential growth, suitable for students in Grades 10-12.

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