Question

A boat is heading towards a lighthouse, whose beacon-light is 104 feet above the water. From point A, the boat’s crew measures the angle of elevation to the beacon 7°, before they draw closer. They measure the angle of elevation a second time from point B at some later time to be 24° Find the distance from point A to point B. Round your answer to the nearest tenth of a foot if necessary.

Answer 1

The distance the boat traveled from point A to point B, after calculating the horizontal distances from the angles of elevation to the lighthouse, is approximately 598.9 feet, which can be rounded to 599 feet.

Step-by-step explanation:

To find the distance the boat traveled from point A to point B, we can use trigonometry and the concept of angle of elevation. The angle of elevation is the angle between the horizontal line and the line of sight looking up at an object. Given the lighthouse beacon-light is 104 feet above the water, we can create two right-angled triangles with the lighthouse as the peak, point A as one base corner with a 7° angle, and point B as another base corner with a 24° angle.

First, we can calculate the distance of the boat from the lighthouse at points A and B using the tangent of the angles of elevation:

• tan(7°) = opposite/adjacent → adjacent_A (distance from A to lighthouse) = opposite/tan(7°) = 104 / tan(7°)
• tan(24°) = opposite/adjacent → adjacent_B (distance from B to lighthouse) = opposite/tan(24°) = 104 / tan(24°)

The distance from point A to B is the difference between the two distances:

distance_AB = adjacent_A - adjacent_B

Plugging in the values and performing the calculations (using a calculator for the tangent values), we find:

adjacent_A = 104 / tan(7°) = 848.8 feet (approximately)

adjacent_B = 104 / tan(24°) = 249.9 feet (approximately)

distance_AB = 848.8 feet - 249.9 feet = 598.9 feet

Therefore, the distance the boat traveled from point A to point B is approximately 598.9 feet, which can be rounded to 599 feet to the nearest tenth of a foot if necessary.

Answer 2

The distance from point A to point B is approximately 29.3 feet.

Step-by-step explanation:

We can solve this problem using trigonometry, specifically the tangent function. Here's how:

1. Set up the diagram:

Imagine a right triangle where:

The lighthouse beacon is at the top vertex (O).

Point A is at the base (bottom left vertex).

Point B is another point on the base (bottom right vertex).

The angle of elevation from point A is 7° (∠AOB).

The angle of elevation from point B is 24° (∠BOH).

The height of the lighthouse beacon above the water is 104 feet (OB).

The distance from point A to the lighthouse is represented by AB (x).

The distance from point B to the lighthouse is represented by BH (y).

2. Use the tangent function:

The tangent function relates the opposite side (height) to the adjacent side (base) in a right triangle. We know the height (OB = 104 feet) and can use the tangent of the angles to find the base lengths (x and y).

For angle ∠AOB:

tan(7°) = OB / AB = 104 / x

For angle ∠BOH:

tan(24°) = OB / BH = 104 / y

3. Solve for the distance between A and B:

We want to find the distance AB. We can use the equations above to set up a proportion:

AB / BH = tan(7°) / tan(24°)

Substituting the values of the tangents:

x / y = 0.1227 / 0.4452

Cross-multiplying and solving for x:

x = (y * 0.1227) / 0.4452

We don't need to solve for y explicitly. We can substitute the value of OB (104 feet) for y:

x = (104 * 0.1227) / 0.4452

x ≈ 29.3 feet

Therefore, the distance from point A to point B is approximately 29.3 feet.