https://file.aiquickdraw.com/mathbot/file/1727033409129mv7013i8.jpg

The problem extracted from the image states:

---

**12) Un cône droit et une sphère ont la même aire totale. Le rayon du cône est le même que le rayon de la sphère, soit de 11,3 dm. Quelle est la mesure de l'apothème du cône ?**

---

### Translation to English:

A right circular cone and a sphere have the same total surface area. The radius of the cone is the same as the radius of the sphere, which is 11.3 dm. What is the length of the slant height (apothème) of the cone?

---

### Step-by-step Solution:

1. **Surface area of the sphere**:
   The formula for the surface area \( A_s \) of a sphere is:
   \[
   A_s = 4\pi r^2
   \]
   where \( r = 11.3 \) dm. So:
   \[
   A_s = 4\pi (11.3)^2
   \]

2. **Surface area of the cone**:
   The total surface area of a right circular cone consists of the lateral surface area and the area of the circular base. The formula for the surface area \( A_c \) of a cone is:
   \[
   A_c = \pi r l + \pi r^2
   \]
   where \( r \) is the radius, and \( l \) is the slant height (apothème).

3. **Equating the surface areas**:
   Since the surface area of the cone is equal to the surface area of the sphere:
   \[
   \pi r l + \pi r^2 = 4\pi r^2
   \]

4. **Solving for the slant height \( l \)**:
   Simplify the equation:
   \[
   \pi r l = 3\pi r^2
   \]
   Canceling \( \pi \) and \( r \) (since \( r \neq 0 \)):
   \[
   l = 3r
   \]
   Substituting \( r = 11.3 \) dm:
   \[
   l = 3 \times 11.3 = 33.9 \, \text{dm}
   \]

Thus, the apothème (slant height) of the cone is **33.9 dm**.

---

Let me know if you'd like more details or have questions! 

### Related Questions:
1. What is the volume of the cone with the same dimensions?
2. How does the volume of the sphere compare to the volume of the cone?
3. What is the lateral surface area of the cone with the given radius and apothème?
4. What would be the height of the cone if the apothème is 33.9 dm and the radius is 11.3 dm?
5. How would the total surface area change if the radius were doubled?

### Tip:
Always remember that the lateral surface area of a cone depends on both the radius and the slant height, while the area of the base is only influenced by the radius!

Un cône droit et une sphère ont la même aire totale. Le rayon du cône est le même que le rayon de la sphère, soit de 11,3 dm. Quelle est la mesure de l'apothème du cône ?

This geometry problem involves finding the slant height of a right circular cone, given that it has the same surface area as a sphere with a radius of 11.3 dm. The solution includes using the surface area formulas for both shapes and solving for the slant height.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation