Question

A boat is stationed nearby a lighthouse that stands on top of a cliff. the base of the boat at sea level makes a 15 degree angle with the tip of the lighthouse. if the top of the lighthouse is 110 meters above sea level, how far out to sea is the boat, to the nearest tenth of a meter?

Answer 1

To calculate the distance from the boat to the lighthouse, we utilize the tangent function with the given angle and height. By dividing the height of the lighthouse by the tangent of the 15-degree angle, we find the boat to be approximately 410.5 meters out to sea.

Step-by-step explanation:

In order to determine how far out to sea the boat is stationed from the lighthouse which stands on top of a 110-meter cliff, we can use trigonometry. This problem can be considered as finding the length of the adjacent side of a right-angled triangle, with the angle given as 15 degrees and the opposite side (the height of the lighthouse) as 110 meters.

To find the distance of the boat to the lighthouse (adjacent side), we can use the tangent function:

tangent(15°) = opposite/adjacent

adjacent = opposite / tangent(15°)

adjacent = 110 meters / tangent(15°)

Using a calculator, we can find that tangent(15°) ≈ 0.2679. Therefore:

adjacent = 110 meters / 0.2679 ≈ 410.5 meters

To the nearest tenth of a meter, the boat is approximately 410.5 meters out to sea from the base of the lighthouse.