A boat is spotted in the water with an angle of depression of 25° from the top of a lighthouse that is 89 feet tall. To the nearest foot, how far away is the boat from the base of the lighthouse
Did you make a sketch?
Looks like
tan 25º = 89/x
x = 89/tan25
= 191
To find the distance between the boat and the base of the lighthouse, we can use trigonometry.
Let's call the distance between the boat and the base of the lighthouse "x" (in feet).
We can use the tangent function, which is defined as the opposite over the adjacent side in a right triangle.
In this case, the opposite side is the height of the lighthouse (89 feet) and the adjacent side is the distance between the boat and the base of the lighthouse (x feet).
The tangent of the angle of depression (25°) is equal to the opposite side divided by the adjacent side:
tan(25°) = 89 / x
To find x, we can rearrange the equation:
x = 89 / tan(25°)
Now we can calculate the value of x:
x = 89 / tan(25°) ≈ 198.65
Therefore, the boat is approximately 198.65 feet away from the base of the lighthouse, rounded to the nearest foot.
To find the distance from the base of the lighthouse to the boat, we can use trigonometry. The angle of depression is the angle formed between the line of sight from the top of the lighthouse to the boat and the horizontal ground.
In this case, we have the height of the lighthouse (89 feet) and the angle of depression (25°). We want to find the distance from the base of the lighthouse to the boat.
Let's call the distance from the base of the lighthouse to the boat "d" (in feet). We can set up a right triangle, with the vertical side being the height of the lighthouse (89 feet) and the horizontal side being the distance from the base of the lighthouse to the boat (d feet).
To find the distance "d," we can use the tangent function, which is the ratio of the opposite side (89 feet) to the adjacent side (d feet):
tan(25°) = 89 / d
To isolate "d," we can rearrange the equation:
d = 89 / tan(25°)
We can now plug this into a calculator to find the value of "d" to the nearest foot. The final result will be the distance from the base of the lighthouse to the boat.