The height of a right circular cone is 23.7 centimeters, and the diameter of the base is 18 centimeters. What angle does the side of the cone make with the base( rounded to at least the nearest tenth)?

To determine the angle that the side of a cone makes with its base, we need to use trigonometric relationships, specifically involving the right triangle formed by the height, radius, and slant height of the cone.

### Steps:

1. **Calculate the radius**:  
   The radius \( r \) is half the diameter of the base.  
   \[
   r = \frac{18}{2} = 9 \, \text{cm}
   \]

2. **Right triangle relationship**:  
   In a right circular cone, the height \( h \), the radius \( r \), and the slant height \( l \) form a right triangle. We can calculate the angle \( \theta \) formed by the side (slant height) and the base using the tangent function:
   \[
   \tan(\theta) = \frac{r}{h}
   \]

3. **Calculate the angle**:
   Substituting the known values:
   \[
   \tan(\theta) = \frac{9}{23.7}
   \]
   \[
   \theta = \tan^{-1}\left(\frac{9}{23.7}\right)
   \]

Let me calculate this value for you.The angle \( \theta \) that the side of the cone makes with the base is approximately \( 20.8^\circ \), rounded to the nearest tenth.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you calculate the slant height of a cone?
2. What is the surface area of this cone?
3. How does the angle change if the cone’s height increases?
4. What is the volume of the cone given the dimensions?
5. How can you calculate the lateral surface area of the cone?

**Tip**: In many problems involving angles, trigonometric ratios like sine, cosine, and tangent are very useful for relating side lengths to angles.

The height of a right circular cone is 23.7 centimeters, and the diameter of the base is 18 centimeters. What angle does the side of the cone make with the base (rounded to at least the nearest tenth)?

Learn how to calculate the angle that the side of a right circular cone makes with its base using trigonometric relationships. Given a cone with a height of 23.7 cm and a base diameter of 18 cm, this solution shows how to apply the tangent function and inverse tangent to solve for the angle, which is approximately 20.8°. Suitable for high school students.

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