Question

A boat is heading towards a lighthouse, whose beacon-light is 142 feet above the water. From point A, the boat’s crew measures the angle of elevation to the beacon, 9 degrees before they draw closer. They measure the angle of elevation a second time from point B at some later time to be 25 degrees Find the distance from point A to point B. Round your answer to the nearest tenth of a foot if necessary.

Answer 1

964.4 feet.

Step by Step explanation:

Let's call the distance from point A to the lighthouse "x", and the height of the boat at point B "h".

Using trigonometry, we can set up two equations based on the angles of elevation measured at points A and B:

tan(9) = 142/x

tan(25) = (142+h)/x

We can rearrange the first equation to solve for x:

x = 142/tan(9)

x ≈ 935.4 feet

Substituting this value into the second equation and solving for h:

tan(25) = (142+h)/935.4

h = 935.4 * tan(25) - 142

h ≈ 377.5 feet

So the height of the boat at point B is approximately 377.5 feet. To find the distance from point A to point B, we can use the Pythagorean theorem:

distance = √(x^2 + h^2)

distance ≈ 964.4 feet

Therefore, the distance from point A to point B is approximately 964.4 feet.

Answer 2

964.4 feet.

Explanation:

Let's call the distance from point A to the lighthouse "x", and the height of the boat at point B "h".

Using trigonometry, we can set up two equations based on the angles of elevation measured at points A and B:

tan(9) = 142/x

tan(25) = (142+h)/x

We can rearrange the first equation to solve for x:

x = 142/tan(9)

x ≈ 935.4 feet

Substituting this value into the second equation and solving for h:

tan(25) = (142+h)/935.4

h = 935.4 * tan(25) - 142

h ≈ 377.5 feet

So the height of the boat at point B is approximately 377.5 feet. To find the distance from point A to point B, we can use the Pythagorean theorem:

distance = √(x^2 + h^2)

distance ≈ 964.4 feet

Therefore, the distance from point A to point B is approximately 964.4 feet.