https://file.aiquickdraw.com/mathbot/file/1727289828345l5rqi04w.jpg

The question is asking for the ratio of the surface areas of two similar cones, where the ratio of their radii and altitudes is given as 2:1.

### Steps to solve:

1. **Ratio of radii and altitudes:**  
   Since the cones are similar, the ratio of their corresponding linear dimensions (such as radii and altitudes) is 2:1.

2. **Surface area of a cone:**  
   The formula for the surface area of a cone is:
   \[
   A = \pi r(r + l)
   \]
   where \(r\) is the radius and \(l\) is the slant height. However, since we are dealing with similar cones, we only need to focus on the relationship between their surface areas.

3. **Surface area scaling law for similar figures:**  
   For two similar 3D shapes, the ratio of their surface areas is the square of the ratio of their corresponding linear dimensions. If the ratio of the radii (and altitudes) is \(2:1\), then the ratio of their surface areas will be:
   \[
   \left(\frac{2}{1}\right)^2 = 4:1
   \]

Therefore, the correct answer is:

\[
\boxed{D. 4:1}
\]

Would you like more details, or do you have any questions?

Here are five related questions for further practice:
1. How does the ratio of the volumes of two similar cones compare to the ratio of their linear dimensions?
2. What would the ratio of surface areas be if the ratio of radii was 3:1?
3. How does the slant height affect the surface area of a cone?
4. Can the surface area ratio be used to calculate the ratio of radii between two cones?
5. What is the formula for the lateral surface area of a cone?

**Tip:** For any two similar 3D shapes, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions.

What is the ratio for the surface areas of the cones shown below, given that they are similar and the ratio of their radii and altitudes is 2:1?

Learn how to find the ratio of surface areas for two similar cones where the ratio of their radii and altitudes is 2:1. This problem covers concepts related to similar shapes, surface area scaling, and geometry, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation