The drawing shows a person looking at a building on top of which an antenna is mounted. The horizontal distance between the person's eyes and the building is d = 83.0 m. In part a the person is looking at the base of the antenna, and his line of sight makes an angle of θ1 = 35.0° with the horizontal. In part b the person is looking at the top of the antenna, and his line of sight makes an angle of θ2 = 39.0° with the horizontal. How tall is the antenna?
Δx=83(tan 39º-tan35º)= =
=83(0.81 -0.70) =9.13 m
To find the height of the antenna, we need to use trigonometry. Let's start solving this step-by-step.
Step 1: Draw a diagram to visualize the problem.
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| | <- Building
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__|__|__
|<- 83.0 m ->|
|_________| <- Person's eyes
Step 2: Label the given angles and distances on the diagram.
θ1
__________|___________
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| |
83.0 m |
| |
| θ2 |
|______________________|
Antenna
Step 3: Use trigonometric functions to establish equations for the problem.
In part a, we have:
angle θ1 = 35.0°
horizontal distance = 83.0 m
In part b, we have:
angle θ2 = 39.0°
horizontal distance = 83.0 m
Step 4: Calculate the heights of the antenna for part a and part b separately.
For part a, let's call the height of the antenna h1.
Using trigonometry, we can write:
tan(θ1) = h1 / 83.0 m
Rearranging the equation:
h1 = tan(θ1) * 83.0 m
For part b, let's call the height of the antenna h2.
Using trigonometry, we can write:
tan(θ2) = h2 / 83.0 m
Rearranging the equation:
h2 = tan(θ2) * 83.0 m
Step 5: Substitute the given values and calculate the heights of the antenna.
Using a calculator, find the values of tan(θ1) and tan(θ2) (tan(35.0°) and tan(39.0°)):
tan(θ1) = 0.7002075382
tan(θ2) = 0.8097840332
Now substitute these values into the equations for h1 and h2:
h1 = 0.7002075382 * 83.0 m
h1 = 58.11 m (rounded to two decimal places)
h2 = 0.8097840332 * 83.0 m
h2 = 67.23 m (rounded to two decimal places)
Step 6: Calculate the height of the antenna by subtracting the heights of part a and part b.
height of antenna = h2 - h1
height of antenna = 67.23 m - 58.11 m
height of antenna = 9.12 m (rounded to two decimal places)
The height of the antenna is approximately 9.12 meters.
To find the height of the antenna, we can use trigonometry. Let's break down the problem into two parts:
Part a: The person is looking at the base of the antenna.
In this case, we can consider the right triangle formed by the person's line of sight, the horizontal distance, and the height of the antenna. We need to find the height of the antenna.
Using the trigonometric function tangent:
tan(theta1) = height / horizontal distance
Rearranging the formula, we get:
height = tan(theta1) * horizontal distance
Substituting the given values:
theta1 = 35.0°
horizontal distance = 83.0 m
Calculating:
height = tan(35.0°) * 83.0 m
Part b: The person is looking at the top of the antenna.
In this case, we can again consider the right triangle formed by the person's line of sight, the horizontal distance, and the height of the antenna. We need to find the total height of the antenna.
Using the same trigonometric function tangent:
tan(theta2) = height + antenna base / horizontal distance
Rearranging the formula, we get:
height + antenna base = tan(theta2) * horizontal distance
Since we already know the antenna base is the height we calculated in part a, we can substitute the value:
height + height = tan(theta2) * horizontal distance
Substituting the given values:
theta2 = 39.0°
horizontal distance = 83.0 m
Calculating:
height + height = tan(39.0°) * 83.0 m
Simplifying the equation:
2 * height = tan(39.0°) * 83.0 m
Solving for height, we get:
height = 0.5 * tan(39.0°) * 83.0 m
Calculating the final result:
height = 0.5 * tan(39.0°) * 83.0 m = 38.2 m
Therefore, the height of the antenna is 38.2 meters.