A lighthouse that rises h = 50 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base, as shown in the figure. A sailor on the deck of a ship sights the top of the lighthouse at an angle of θ = 28.0° above the horizontal. If the sailor's eye level is 14 ft above the water, how far is the ship from the rocks?

Question

A lighthouse that rises h = 50 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base, as shown in the figure. A sailor on the deck of a ship sights the top of the lighthouse at an angle of θ = 28.0° above the horizontal. If the sailor's eye level is 14 ft above the water, how far is the ship from the rocks?

Answer 1

horizonal distance total=rock distance+19ft

height up to top= 50-14

Tan28=heightuptoTop/horizontal distance

tan28deg=(36/(d+14))

solve for d

Answer 2

To find the distance from the ship to the rocks, we can use the concept of trigonometry and create a right triangle. Here's how we can do it step by step:

Step 1: Draw a diagram:
Make a diagram with the lighthouse, the rocky cliff, the sailor on the ship, and marked heights and angles. Label the given lengths and heights.

Step 2: Identify the relevant trigonometric function:
Since we are given the angle of elevation (θ) and we want to find the distance from the ship to the rocks, we need to use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.

Step 3: Identify the opposite and adjacent sides:
In our triangle, the opposite side is the height of the lighthouse (h) plus the sailor's eye level (14 ft). So, the opposite side (O) will be h + 14.

The adjacent side is the distance from the ship to the rocks, which we need to find. So, let's label that side as the distance D.

Step 4: Set up the equation using the tangent function:
Using the tangent function, we can write the equation as follows:
tan(θ) = O / A

Since we know the value of θ and O (h + 14), our equation becomes:
tan(28°) = (h + 14) / D

Step 5: Solve for D:
To find D, we will rearrange the equation and solve for it. Multiply both sides of the equation by D:
D * tan(28°) = h + 14

Next, isolate D by dividing both sides by tan(28°):
D = (h + 14) / tan(28°)

Step 6: Substitute the given values and calculate D:
Substitute the given values: h = 50 ft and tan(28°) ≈ 0.531.
D = (50 + 14) / 0.531
D ≈ 93.44 ft

Therefore, the ship is approximately 93.44 ft from the rocks.