https://file.aiquickdraw.com/mathbot/file/1727390953486g5cwxcgi.jpg

If 
∠
A
C
D
=
15
o
 and 
∠
A
C
B
=
30
o
,
 what is the length of line 
¯¯¯¯¯¯
D
B
 to the nearest tenth of a meter.

To solve this problem and find the length of line \( DB \), we can break it down using the given angles and the geometry of the triangles.

### Step-by-step approach:
1. **Identify Triangles**:
   - We have two right-angled triangles: \( \triangle ACD \) and \( \triangle ACB \).
   - We are given that \( \angle ACD = 15^\circ \) and \( \angle ACB = 30^\circ \).
   - The length of \( AC \) is 44.1 m.

2. **Using Trigonometry in Right Triangles**:
   - We will use the sine and cosine rules in the two right triangles \( \triangle ACD \) and \( \triangle ACB \).

3. **Find the Length of \( AD \)**:
   - In \( \triangle ACD \), the angle at \( C \) is \( 15^\circ \), and the adjacent side is \( AC = 44.1 \) m.
   - Using the cosine of \( 15^\circ \), we can find \( AD \).

   \[
   AD = AC \cdot \cos(15^\circ) = 44.1 \cdot \cos(15^\circ)
   \]

4. **Find the Length of \( CD \)**:
   - Similarly, in the same triangle \( ACD \), we can use the sine of \( 15^\circ \) to find \( CD \):

   \[
   CD = AC \cdot \sin(15^\circ) = 44.1 \cdot \sin(15^\circ)
   \]

5. **Find the Length of \( BD \)**:
   - Now, in \( \triangle BCD \), we know \( \angle BCD = 15^\circ \) and \( \angle DCB = 30^\circ \), allowing us to apply the law of sines or use trigonometric relationships to find \( DB \).

I'll calculate each of these steps for you now.We have calculated the following:

- \( AD \approx 42.6 \, \text{m} \)
- \( CD \approx 11.4 \, \text{m} \)

Next, we can use the triangle \( BCD \) to find \( DB \). We will apply the Law of Sines, since we have angle measures \( \angle BCD = 15^\circ \), \( \angle BCB = 30^\circ \), and the side \( CD \). I'll proceed with these calculations.The length of line \( DB \) is approximately \( 22.1 \, \text{m} \).

Would you like more details on any of these steps?

Here are 5 related questions for further exploration:
1. How is the Law of Sines applied in non-right triangles?
2. Can we solve the same problem using the Law of Cosines?
3. What other methods could be used to find unknown lengths in triangles?
4. How would the problem change if different angles were given?
5. Can you explain how trigonometric functions relate to right triangles?

**Tip**: When dealing with right triangles, always consider which trigonometric function (sine, cosine, or tangent) fits best based on the known sides and angles.

If ∠ACD = 15° and ∠ACB = 30°, what is the length of line DB to the nearest tenth of a meter?

Learn how to calculate the length of line DB in a trigonometric problem involving two right-angled triangles. Using the Law of Sines and the given angles ∠ACD = 15° and ∠ACB = 30°, find the length of DB with a detailed step-by-step solution. This is suitable for students in Grades 10-12.

Math Problem Statement

If ∠ A C D

15 o and ∠ A C B

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation