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In the figure, a person standing at point A notices that the angle of elevation to the top of the antenna is 48° 30'. A second person standing 41.0 feet farther from the antenna than the person at A finds the angle of elevation to the top of the antenna to be 42° 10'. How far is the person at A from the base of the antenna? (Round your answer to the nearest whole number.)  

This is a typical trigonometry problem involving two right triangles, each formed by the observer's line of sight to the top of the antenna. The information given allows us to use the tangent function (since we have angles and need to relate the height and distance).

Let's denote:
- \( h \) as the height of the antenna.
- \( x \) as the distance from the person at point A to the base of the antenna.
- The second person is 41 feet farther away than the person at point A, so the second distance is \( x + 41 \).

For the first person (at point A):
\[
\tan(48^\circ 30') = \frac{h}{x}
\]

For the second person (41 feet farther):
\[
\tan(42^\circ 10') = \frac{h}{x + 41}
\]

We can solve this system of equations to find \( x \).

### Steps:
1. Use the tangent of both angles to express \( h \) in terms of \( x \) in both equations.
2. Equate the two expressions for \( h \) to eliminate \( h \) and solve for \( x \).

Let me calculate that now.The person at point A is approximately 165 feet from the base of the antenna.

Would you like any further details on how this was calculated or have any questions? Here are some related questions:

1. How is the tangent function applied in real-world scenarios like this one?
2. What would the height of the antenna be in this case?
3. How do different angles of elevation impact the calculations in trigonometry?
4. What if the distances or angles were measured in meters—how would the solution differ?
5. Can this problem be solved graphically as well as algebraically?

**Tip:** When dealing with angles in degrees and minutes, converting to decimal degrees helps simplify trigonometric calculations.

In the figure, a person standing at point A notices that the angle of elevation to the top of the antenna is 48° 30'. A second person standing 41.0 feet farther from the antenna than the person at A finds the angle of elevation to the top of the antenna to be 42° 10'. How far is the person at A from the base of the antenna? (Round your answer to the nearest whole number.)

Solve a trigonometry problem involving angles of elevation and distance to an antenna. Learn how to use the tangent function to find distances in right triangles, with a step-by-step solution. Suitable for high school students in Grades 9-12.

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