11. A forest ranger needs to estimate the height of a sequoia tree in the forest. At one point, the angle of elevation to the top of the tree is 15°12'. The ranger walks forward 35 feet from this point toward the tree and remeasures the angle of elevation to be 25°. To the nearest foot, how tall is the tree? A complete solution must include a properly labelled diagram.
11. A forest ranger needs to estimate the height of a sequoia tree in the forest. At one point, the angle of elevation to the top of the tree is 15°12'. The ranger walks forward 35 feet from this point toward the tree and remeasures the angle of elevation to be 25°. To the nearest foot, how tall is the tree? A complete solution must include a properly labelled diagram.
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Chapter1: Trigonometry
Section1.8: Applications And Models
Problem 4ECP: From the time a small airplane is 100 feet high and 1600 ground feet from its landing runway, the...
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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.
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![### Problem 11:
A forest ranger needs to estimate the height of a sequoia tree in the forest. At one point, the angle of elevation to the top of the tree is \(15^\circ 12'\). The ranger walks forward 35 feet from this point toward the tree and remeasures the angle of elevation to be \(25^\circ\). To the nearest foot, how tall is the tree? A complete solution must include a properly labeled diagram.
**Solution Explanation:**
1. **Initial Setup:**
- Let \( h \) be the height of the tree we need to find.
- The ranger initially stands at point \( A \) and measures the angle of elevation \( \theta_1 = 15^\circ 12' \).
- After walking 35 feet towards the tree, the ranger is at point \( B \) and measures a new angle of elevation \( \theta_2 = 25^\circ \).
2. **Forming Right Triangles:**
- Let \( D \) be the base of the tree, directly below the top of the tree.
- The distance \( AD = x \) and \( BD = x - 35 \).
3. **Using Trigonometric Ratios:**
- From the first observation point \( A \):
\[
\tan(15^\circ 12') = \frac{h}{x}
\]
- From the second observation point \( B \):
\[
\tan(25^\circ) = \frac{h}{x - 35}
\]
4. **Solving for \( h \):**
- Calculate \( \tan(15^\circ 12') \approx 0.271 \) and \( \tan(25^\circ) \approx 0.466 \).
- Set up the equations:
\[
h = x \cdot 0.271
\]
\[
h = (x - 35) \cdot 0.466
\]
- Equate the two expressions for \( h \):
\[
x \cdot 0.271 = (x - 35) \cdot 0.466
\]
\[
0.271x = 0.466x - 16.31
\]
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Transcribed Image Text:### Problem 11:
A forest ranger needs to estimate the height of a sequoia tree in the forest. At one point, the angle of elevation to the top of the tree is \(15^\circ 12'\). The ranger walks forward 35 feet from this point toward the tree and remeasures the angle of elevation to be \(25^\circ\). To the nearest foot, how tall is the tree? A complete solution must include a properly labeled diagram.
**Solution Explanation:**
1. **Initial Setup:**
- Let \( h \) be the height of the tree we need to find.
- The ranger initially stands at point \( A \) and measures the angle of elevation \( \theta_1 = 15^\circ 12' \).
- After walking 35 feet towards the tree, the ranger is at point \( B \) and measures a new angle of elevation \( \theta_2 = 25^\circ \).
2. **Forming Right Triangles:**
- Let \( D \) be the base of the tree, directly below the top of the tree.
- The distance \( AD = x \) and \( BD = x - 35 \).
3. **Using Trigonometric Ratios:**
- From the first observation point \( A \):
\[
\tan(15^\circ 12') = \frac{h}{x}
\]
- From the second observation point \( B \):
\[
\tan(25^\circ) = \frac{h}{x - 35}
\]
4. **Solving for \( h \):**
- Calculate \( \tan(15^\circ 12') \approx 0.271 \) and \( \tan(25^\circ) \approx 0.466 \).
- Set up the equations:
\[
h = x \cdot 0.271
\]
\[
h = (x - 35) \cdot 0.466
\]
- Equate the two expressions for \( h \):
\[
x \cdot 0.271 = (x - 35) \cdot 0.466
\]
\[
0.271x = 0.466x - 16.31
\]
\[
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