A lighthouse that rises 46.0 ft above the surface of the water sits on a rocky cliff that extends 19 ft out into the ocean from the base of the lighthouse. A sailor on the forward deck of a ship sights the top of the lighthouse at an angle of 25.0 degrees above the horizontal. If the sailors eye level is 14 ft above the water, how many ft is the ship from the rocks?
To find the distance from the ship to the rocks, we can use trigonometry. Let's break down the problem into smaller parts:
1. Find the height of the lighthouse from the horizontal line (the sailor's eye level):
The sailor's eye level is 14 ft above the water, and the lighthouse rises 46 ft above the water. Therefore, the total height from the sailor's eye level to the top of the lighthouse is 46 + 14 = 60 ft.
2. Find the distance from the lighthouse to the rocks:
The rocky cliff extends 19 ft into the ocean from the base of the lighthouse.
3. Find the distance from the ship to the rocks (unknown):
Let's denote this distance as "x."
Now, we can set up a right triangle using the given information. The vertical leg of the triangle represents the height of the lighthouse from the horizontal line, and the horizontal leg represents the distance from the ship to the rocks.
Using the trigonometric function tangent (tan), we can write the equation:
tan(25) = Height of lighthouse from horizontal line / Distance from ship to the rocks
Using the given values, we substitute in the equation:
tan(25) = 60 / x
Now, we solve for x by isolating it:
x = 60 / tan(25)
Using a calculator, we can find that:
x ≈ 123.2 ft
Therefore, the ship is approximately 123.2 ft from the rocks.
To solve this problem, we can use trigonometry and create a right triangle.
Let's label the relevant lengths in the problem:
- The height of the lighthouse above the water surface is the opposite side of the angle, which is 46.0 ft.
- The distance from the front of the ship to the lighthouse base is the adjacent side of the angle, which is what we need to find.
- The sailor's eye level above the water is the height of the triangle, which is 14 ft.
- The distance from the lighthouse base to the water (rocky cliff) is the hypotenuse, which is the sum of the lighthouse's height and the rocky cliff's extension, so it is 46.0 ft + 19.0 ft = 65.0 ft.
We can use the tangent function to find the value of the adjacent side, which represents the distance we're looking for. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle.
So we have the equation:
tan(angle) = opposite/adjacent
Plugging in the values we know:
tan(25.0 degrees) = 46.0 ft/adjacent
To solve for the adjacent side, we can rearrange the equation:
adjacent = opposite/tan(angle)
Now we can substitute in the values:
adjacent = 46.0 ft/tan(25.0 degrees)
Using a scientific calculator, we can find:
adjacent ≈ 105.30 ft
Therefore, the ship is approximately 105.30 ft from the rocks.