The angle of depression from the top of a lighthouse to the base of a boat out at sea is 10°. If the lighthouse is at the edge of land and is 82 meters tall, approximately how far out at sea is the boat?
Question
The angle of depression from the top of a lighthouse to the base of a boat out at sea is 10°. If the lighthouse is at the edge of land and is 82 meters tall, approximately how far out at sea is the boat?
Solution
To solve this problem, we can use trigonometry. The angle of depression is the angle between the line of sight from the top of the lighthouse to the boat and the horizontal line.
Step 1: Draw a diagram to visualize the problem. Label the height of the lighthouse as 82 meters and the angle of depression as 10°.
Step 2: Use the tangent function to find the distance from the lighthouse to the boat. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse and the adjacent side is the distance from the lighthouse to the boat.
Step 3: Set up the equation: tan(10°) = 82/distance. Rearrange the equation to solve for the distance: distance = 82/tan(10°).
Step 4: Use a calculator to find the tangent of 10°. The tangent of 10° is approximately 0.176.
Step 5: Substitute the value of the tangent into the equation: distance = 82/0.176.
Step 6: Calculate the distance: distance ≈ 465.91 meters.
Therefore, the boat is approximately 465.91 meters out at sea from the lighthouse.
Similar Questions
A yacht is anchored 90 feet offshore from the base of a lighthouse. The angle of elevation from the boat to the top of the lighthouse is 26 degrees. The distance between the yacht and the top of the lighthouse is about 100 feet. Calculate the height of the lighthouse? (Round your answer to the nearest whole number.)
A boat is heading towards a lighthouse, whose beacon-light is 140 feet above the water. The boat’s crew measures the angle of elevation to the beacon, 10degrees ∘ . What is the ship’s horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest tenth of a foot if necessary.
A ship is 100 meters from a lighthouse. The angle of depression from the top of the lighthouse to the ship is 20°. What is the distance of the line of sight from the ship to the top of the lighthouse?
An observer in a lighthouse is 45 meters above the surface of the water. The observer sighted two ships due east. The angle of depression to ship A is 51 degrees and the angle of depression to ship B is 24 degrees. Which ship is closer to the base of the lighthouse?
The lighthouse stands 10 meters tall.Boat A is directly south of the lighthouse.The angle of depression from the lighthouse's top to Boat A is 30°.Boat B is 20 meters north and 20 meters east of the lighthouse.An island creates a 130° angle between Boat A and Boat B.The angle at Boat B between Boat A and the island is 30°.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.