To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75° from the horizontal. An observer D = 600 m away measures the angle of elevation to the spot of light to be 45°. Find the height h of the cloud cover. (Round your answer to the nearest meter.)
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To find the height of the cloud cover, we can use trigonometry. Let's break down the problem step by step:
1. Draw a diagram: Draw a horizontal line to represent the ground. At one end of the line, draw a vertical line to represent the cloud cover. Now, draw a line from the other end of the ground line at an angle of 75° to represent the spotlight. Finally, draw a line from the observer's position to the spot of light at an angle of 45° to the ground line.
2. Label the diagram: Label the vertical line as "h" to represent the height of the cloud cover. Label the horizontal line as "600 m" to represent the distance between the observer and the spotlight.
3. Identify the trigonometric ratios: In this problem, we can use the tangent function to relate the angle of elevation and the height of the cloud cover. Recall that the tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
4. Set up the trigonometric equation: In this case, we have: tan(45°) = h / 600 m.
5. Solve for the height: Rearrange the equation to solve for h: h = tan(45°) * 600 m.
6. Use a calculator: Calculate the height h using a scientific calculator or an online trigonometric calculator. The value of tan(45°) is 1, so the equation simplifies to h = 1 * 600 m = 600 m.
Therefore, the height of the cloud cover is approximately 600 meters.
use the Law of Sines to find the distance from one of the people to the spot
then use trig to find the height of the spot
the angle at the spot is
... 180º - 75º - 45º = 60º
600 m / sin(60º) = x / sin(45º)
... x is the distance from the observer to the spot
h = x * sin(45º)