### Problem ContextA man standing near a radio station antenna observes that the angle of elevation to the top of the antenna is $ b = 67^\circ $. He then walks 100 feet further away and observes that the angle of elevation to the top of the antenna is $ a = 45^\circ $. ### TaskFind the height of the antenna to the nearest foot. **Hint:** Find $ x $ first. You must show work on a separate sheet of paper to receive credit.### Diagram ExplanationIn the diagram, there are two people represented as small figures: - The first person is standing directly adjacent to the antenna and their angle of elevation is $ b = 67^\circ $.- The second person is standing 100 feet further away from the first person (shown by a flat line segment labeled "100 ft") and their angle of elevation is $ a = 45^\circ $.The diagram shows two right triangles with the antenna labeled as height $ h $:- The bottom portion of the first triangle, split horizontally from the first person to the base of the antenna, is $ x $.- The bottom portion of the second triangle includes the segment split horizontally from the second person to the base of the antenna, which totals $ x + 100 $ since the second person moved 100 feet further away.### ConclusionTo solve for $ h $, first find $ x $. Then use the right triangle trigonometric properties of sine or tangent functions to solve for $ h $.The goal is to calculate the height ($ h $) of the radio station antenna in feet.\[ h = \_\_\_\_\_\_\_ ft \]

Follow the steps below to find the answer.

A man standing near a radio station antenna observes that the angle of elevation to the top of the antenna is b = 67°. He then walks 100 feet further away and observes that the angle of elevation to the top of the antenna is a = 45°. Find the height of the antenna to the nearest foot. (Hint: Find x first.) You must show work on a separate sheet of paper to receive credit. 100 ft h = ft

A man standing near a radio station antenna observes that the angle of elevation to the top of the antenna is b = 67°. He then walks 100 feet further away and observes that the angle of elevation to the top of the antenna is a = 45°. Find the height of the antenna to the nearest foot. (Hint: Find x first.) You must show work on a separate sheet of paper to receive credit. 100 ft h = ft

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.1: Angles

Problem 27E

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Concept explainers

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

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### Problem Context

A man standing near a radio station antenna observes that the angle of elevation to the top of the antenna is $ b = 67^\circ $. He then walks 100 feet further away and observes that the angle of elevation to the top of the antenna is $ a = 45^\circ $.

### Task

Find the height of the antenna to the nearest foot.

**Hint:** Find $ x $ first. You must show work on a separate sheet of paper to receive credit.

### Diagram Explanation

In the diagram, there are two people represented as small figures:
- The first person is standing directly adjacent to the antenna and their angle of elevation is $ b = 67^\circ $.
- The second person is standing 100 feet further away from the first person (shown by a flat line segment labeled "100 ft") and their angle of elevation is $ a = 45^\circ $.

The diagram shows two right triangles with the antenna labeled as height $ h $:
- The bottom portion of the first triangle, split horizontally from the first person to the base of the antenna, is $ x $.
- The bottom portion of the second triangle includes the segment split horizontally from the second person to the base of the antenna, which totals $ x + 100 $ since the second person moved 100 feet further away.

### Conclusion

To solve for $ h $, first find $ x $. Then use the right triangle trigonometric properties of sine or tangent functions to solve for $ h $.

The goal is to calculate the height ($ h $) of the radio station antenna in feet.

\[ h = \_\_\_\_\_\_\_ ft \]

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