A cylinder of height 2x is circumscribed by a sphere of radius 2x such that the circular ends of the cylinder are two small circles on the sphere. What is the ratio of the curved surface area of the cylinder to the surface area of the sphere?

Options

A. `sqrt3:4`
B. `sqrt3:3`
C. `sqrt3:2`
D. `sqrt3:1`

Answer

To solve the problem, we need to find the ratio of the curved surface area of a cylinder to the surface area of a sphere that circumscribes it. Let's break it down step by step.

### Step 1: Identify the dimensions of the cylinder and sphere
- The height of the cylinder (h) is given as \(2x\).
- The radius of the sphere (R) is given as \(2x\).

### Step 2: Determine the radius of the cylinder
Since the cylinder is circumscribed by the sphere, we can use the Pythagorean theorem to find the radius of the cylinder (r). The center of the cylinder is at the center of the sphere, and the height of the cylinder is divided equally above and below the center.

- The distance from the center of the sphere to the top or bottom of the cylinder is \(x\) (half of the height).
- The radius of the sphere is \(2x\).

Using the Pythagorean theorem:
\[ R^2 = h^2 + r^2 \]
Substituting the known values:
\[ (2x)^2 = (x)^2 + r^2 \]
\[ 4x^2 = x^2 + r^2 \]
\[ r^2 = 4x^2 - x^2 = 3x^2 \]
\[ r = \sqrt{3}x \]

### Step 3: Calculate the curved surface area of the cylinder
The formula for the curved surface area (CSA) of a cylinder is:
\[ \text{CSA} = 2\pi rh \]
Substituting the values we have:
\[ \text{CSA} = 2\pi (\sqrt{3}x)(2x) = 4\pi\sqrt{3}x^2 \]

### Step 4: Calculate the surface area of the sphere
The formula for the surface area (SA) of a sphere is:
\[ \text{SA} = 4\pi R^2 \]
Substituting the radius of the sphere:
\[ \text{SA} = 4\pi (2x)^2 = 4\pi (4x^2) = 16\pi x^2 \]

### Step 5: Find the ratio of the curved surface area of the cylinder to the surface area of the sphere
Now we can find the ratio:
\[ \text{Ratio} = \frac{\text{CSA of cylinder}}{\text{SA of sphere}} = \frac{4\pi\sqrt{3}x^2}{16\pi x^2} \]
Cancelling out common terms:
\[ \text{Ratio} = \frac{4\sqrt{3}}{16} = \frac{\sqrt{3}}{4} \]

### Final Answer
The ratio of the curved surface area of the cylinder to the surface area of the sphere is:
\[ \frac{\sqrt{3}}{4} \]