a person in a boat sights the top of a lighthouse at an angle of elevation of 20 degrees. if the lighthouse is 40 feet tall find the distance from the base of the lighthouse to the boat.
length oflighthouse=4om
angle of depression=20
TAN 20=0.363
put the value and get the answer
answer is approx 108.xx
To find the distance from the base of the lighthouse to the boat, we can use trigonometry.
Let's assume that the distance from the boat to the base of the lighthouse is represented by 'x'.
In this situation, we have a right triangle. The height of the lighthouse is the 'opposite' side, and the distance from the boat to the lighthouse is the 'adjacent' side.
We are given an angle of elevation of 20 degrees, which means that the angle between the horizontal line and the line of sight from the boat to the top of the lighthouse is 20 degrees. Therefore, the angle between the line of sight and the vertical line is 90 - 20 = 70 degrees.
Now, we can use the tangent function to find the distance 'x':
tan(70 degrees) = opposite/adjacent
tan(70 degrees) = 40/x
To isolate 'x', we can cross multiply and solve for 'x':
x * tan(70 degrees) = 40
x = 40 / tan(70 degrees)
Using a scientific calculator, we can find that tan(70 degrees) ≈ 2.74747741946.
Therefore,
x = 40 / 2.74747741946
x ≈ 14.55 feet
Thus, the distance from the base of the lighthouse to the boat is approximately 14.55 feet.
To find the distance from the base of the lighthouse to the boat, we can use the concept of trigonometry. In this case, we can use the tangent function since we have the angle of elevation.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (40 feet), and the adjacent side is the distance we are trying to find.
Let's call the distance from the base of the lighthouse to the boat "d".
Therefore, tan(20°) = opposite/adjacent
=> tan(20°) = 40/d
To solve for "d", we can rearrange the equation as follows:
d = 40 / tan(20°)
Now, we can calculate the value of "d" using a scientific calculator or a calculator with trigonometric functions:
d ≈ 115.9 feet
So, the distance from the base of the lighthouse to the boat is approximately 115.9 feet.