3. The angle of elevation of the top of a building from point A on the ground is 24.2'. From point B which is 44.5 feet closer to the building, the angle of elevation is 38.1'. Find the height of the building to the nearest foot.
3. The angle of elevation of the top of a building from point A on the ground is 24.2'. From point B which is 44.5 feet closer to the building, the angle of elevation is 38.1'. Find the height of the building to the nearest foot.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
![### Problem Statement
**Question 3:**
The angle of elevation of the top of a building from point A on the ground is 24.2°. From point B, which is 44.5 feet closer to the building, the angle of elevation is 38.1°. Find the height of the building to the nearest foot.
**Solution Approach:**
This problem involves solving for the height of a building using trigonometric principles, specifically involving angles of elevation and the concept of tangent in a right triangle.
1. **Define the Problem:**
- Let the height of the building be \( h \).
- Let the distance from point A to the base of the building be \( d \).
- Thus, the distance from point B to the base of the building is \( d - 44.5 \).
2. **Apply Trigonometric Functions:**
- From point A, the tangent of the angle of elevation is:
\[
\tan(24.2^\circ) = \frac{h}{d}
\]
- From point B, the tangent of the angle of elevation is:
\[
\tan(38.1^\circ) = \frac{h}{d - 44.5}
\]
3. **Set Up Equations:**
- From point A:
\[
h = d \cdot \tan(24.2^\circ)
\]
- From point B:
\[
h = (d - 44.5) \cdot \tan(38.1^\circ)
\]
4. **Equate the Height Equations:**
\[
d \cdot \tan(24.2^\circ) = (d - 44.5) \cdot \tan(38.1^\circ)
\]
5. **Solve for \( d \):**
\[
d \cdot \tan(24.2^\circ) = d \cdot \tan(38.1^\circ) - 44.5 \cdot \tan(38.1^\circ)
\]
\[
d \cdot (\tan(24.2^\circ) - \tan(38.1^\circ)) = -44.5 \cdot \tan(38.1^\circ)
\]
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Transcribed Image Text:### Problem Statement
**Question 3:**
The angle of elevation of the top of a building from point A on the ground is 24.2°. From point B, which is 44.5 feet closer to the building, the angle of elevation is 38.1°. Find the height of the building to the nearest foot.
**Solution Approach:**
This problem involves solving for the height of a building using trigonometric principles, specifically involving angles of elevation and the concept of tangent in a right triangle.
1. **Define the Problem:**
- Let the height of the building be \( h \).
- Let the distance from point A to the base of the building be \( d \).
- Thus, the distance from point B to the base of the building is \( d - 44.5 \).
2. **Apply Trigonometric Functions:**
- From point A, the tangent of the angle of elevation is:
\[
\tan(24.2^\circ) = \frac{h}{d}
\]
- From point B, the tangent of the angle of elevation is:
\[
\tan(38.1^\circ) = \frac{h}{d - 44.5}
\]
3. **Set Up Equations:**
- From point A:
\[
h = d \cdot \tan(24.2^\circ)
\]
- From point B:
\[
h = (d - 44.5) \cdot \tan(38.1^\circ)
\]
4. **Equate the Height Equations:**
\[
d \cdot \tan(24.2^\circ) = (d - 44.5) \cdot \tan(38.1^\circ)
\]
5. **Solve for \( d \):**
\[
d \cdot \tan(24.2^\circ) = d \cdot \tan(38.1^\circ) - 44.5 \cdot \tan(38.1^\circ)
\]
\[
d \cdot (\tan(24.2^\circ) - \tan(38.1^\circ)) = -44.5 \cdot \tan(38.1^\circ)
\]
\[
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