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%.

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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.