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.)
Can you please give the final answer in the nearest meter with it bc I keep getting it wrong when i try to
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To find the height of the cloud cover, we can use trigonometry. Let's denote the height of the cloud cover as h.
In the given problem, we have two angles: the angle the spotlight shines upward (75°) and the angle of elevation measured by the observer (45°).
Since we have a right triangle, we can use the tangent function to relate the angles and distances. Specifically, we can use the tangent of the angle of elevation to find the height of the cloud cover.
tan(45°) = h / 600
To solve for h, we can rearrange the equation:
h = tan(45°) * 600
Using a calculator, we find:
h ≈ 600 meters
Therefore, the height of the cloud cover is approximately 600 meters.
To find the height of the cloud cover, we can use trigonometry. Let's break down the problem into steps and calculate each part.
1. Draw a diagram: Sketch a right triangle representing the situation described in the problem. Label the vertical side as the height h, the horizontal side as the distance D, the angle of elevation as θ, and the angle at the base as 90°.
2. Identify the known values: Given in the problem, we have:
- The angle of elevation θ = 45°
- The angle of the spotlight from the horizontal α = 75°
- The distance from the observer to the spot of light D = 600 m
3. Determine the trigonometric relationship: In this scenario, we can use the tangent function, which relates the angle of elevation and the ratio of the opposite side (height h) to the adjacent side (distance D). It can be written as:
tan(θ) = h / D
4. Solve for the height: Rearrange the equation to solve for h:
h = D * tan(θ)
Plug in the known values:
h = 600 * tan(45°)
5. Calculate the height h: Using a calculator, find the tan(45°) and calculate the height.
h ≈ 600 * 1 = 600 meters
Therefore, the height of the cloud cover is approximately 600 meters.