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The problem presented is about a circular frustum (圆台) and a sphere (球), where:

1. The radius of the upper base of the frustum is 1.
2. The radius of the lower base of the frustum is 3.
3. The sphere is tangent to both bases and the lateral surface of the frustum.

The task is to find the ratio of the lateral surface area of the frustum to the surface area of the sphere.

### Approach:

#### 1. Surface Area of the Sphere:
The formula for the surface area of a sphere is:
\[
A_{\text{sphere}} = 4\pi r_{\text{sphere}}^2
\]
where \( r_{\text{sphere}} \) is the radius of the sphere.

#### 2. Lateral Surface Area of the Frustum:
The lateral surface area \( A_{\text{lateral}} \) of a circular frustum is given by:
\[
A_{\text{lateral}} = \pi (r_1 + r_2) l
\]
where:
- \( r_1 = 1 \) is the radius of the upper base.
- \( r_2 = 3 \) is the radius of the lower base.
- \( l \) is the slant height, which can be found using the Pythagorean theorem.

#### 3. Ratio of Surface Areas:
The required ratio is:
\[
\text{Ratio} = \frac{A_{\text{lateral}}}{A_{\text{sphere}}}
\]

Let me calculate the required values.The ratio of the lateral surface area of the frustum to the surface area of the sphere is approximately \( 2.24 \).

Do you have any questions or would you like more details? Here are 5 related questions to further explore this topic:

1. How is the slant height of a frustum calculated in general?
2. What would happen to the ratio if the radius of the lower base were increased?
3. How does the surface area of the frustum change if the height remains constant but the radius of the upper base changes?
4. Can you derive the volume of the frustum based on its dimensions?
5. How would this problem change if the frustum and sphere were not tangent?

**Tip:** In problems involving frustums and spheres, visualizing the geometric relationships can greatly help in understanding how the areas relate to each other.

已知圆台上底面半径为1，下底面半径为3，球与圆台的两个底面和侧面均相切，则该圆台的侧面积与球的表面积之比为（）

Learn how to calculate the ratio of the lateral surface area of a circular frustum with upper and lower radii of 1 and 3, respectively, to the surface area of a sphere that is tangent to both bases and the lateral surface. Suitable for students in Grades 10-12.

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