a frustum with bottom dia 3 mtre,vertical axial length 19.756metre and the size of the frustum is increasing when going to top radious. top radious makes 82degree inner angle with the slanting height find the slanting height and top diametre?

To solve this problem, we need to find the slant height (\( l \)) and the top diameter (\( D_{\text{top}} \)) of the frustum.

### Given:
1. Bottom diameter (\( D_{\text{bottom}} \)) = 3 meters
2. Vertical axial length (\( h \)) = 19.756 meters
3. Angle at the top with the slant height (\( \theta \)) = 82 degrees

### Steps:

1. **Calculate the bottom radius**:
   \[
   r_{\text{bottom}} = \frac{D_{\text{bottom}}}{2} = \frac{3}{2} = 1.5 \text{ meters}
   \]

2. **Use trigonometry to find the slant height**:
   The vertical height (\( h \)) and the angle (\( \theta \)) can be related to the slant height (\( l \)) and the difference in radii between the top and bottom.

   The relationship between the vertical height, slant height, and the angle is given by:
   \[
   \cos(\theta) = \frac{h}{l}
   \]
   So, 
   \[
   l = \frac{h}{\cos(\theta)} = \frac{19.756}{\cos(82^\circ)}
   \]

3. **Find the top radius**:
   The top radius \( r_{\text{top}} \) can be calculated using:
   \[
   r_{\text{top}} = r_{\text{bottom}} + l \cdot \sin(\theta)
   \]

4. **Calculate the top diameter**:
   The top diameter \( D_{\text{top}} \) is twice the top radius:
   \[
   D_{\text{top}} = 2 \times r_{\text{top}}
   \]

Let's calculate these values.### Results:

1. **Slant height (\( l \))**: \( 141.95 \) meters
2. **Top diameter (\( D_{\text{top}} \))**: \( 284.14 \) meters

These are the required dimensions of the frustum.

Would you like further details or have any questions?

### Relative Questions:
1. How would you calculate the volume of this frustum?
2. What is the surface area of this frustum, including both top and bottom surfaces?
3. How does the slant height affect the shape of the frustum?
4. What is the significance of the angle with the slant height in this context?
5. How can this method be applied to a conical frustum with different parameters?

### Tip:
Always ensure that angles are converted to radians when using trigonometric functions in calculations.

Learn how to calculate the slant height and top diameter of a frustum with given dimensions and angle. Step-by-step solution using trigonometry and geometry concepts. Suitable for advanced high school students and anyone interested in geometry problems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation