Question

A boat is heading towards a lighthouse, whose beacon-light is 111 feet above the water. From point A, the boat’s crew measures the angle of elevation to the beacon, 6∘ before they draw closer. They measure the angle of elevation a second time from point B at some later time to be 13∘ . Find the distance from point A to point B. Round your answer to the nearest tenth of a foot if necessary.

Answer 1

By applying trigonometry and using tangent functions with the angles of elevation, we calculate the distances from the boat to the lighthouse at points A and B and then find the difference to obtain the distance between A and B.

Step-by-step explanation:

To find the distance from point A to point B as the boat approaches a lighthouse, we can use trigonometry. The lighthouse beacon's light is 111 feet above the water. The angles of elevation from points A and B are given as 6° and 13° respectively. To solve this, we use the tangent function which relates the angle of elevation to the opposite side (height of the lighthouse) and the adjacent side (distance from the lighthouse).

Let x be the distance of the boat from the lighthouse at point A and y be the distance from the lighthouse at point B. The distance between A and B is then x - y.

Using the tangent function for each point, we have:

• tan(6°) = 111/x
• tan(13°) = 111/y

Solving for x and y, we get:

• x = 111 / tan(6°)
• y = 111 / tan(13°)

Calculating these values and subtracting y from x gives us the distance from A to B.

It is important to round the answer to the nearest tenth of a foot, as requested by the student.