John stands on top of a little lighthouse looking out at a nearby tall lighthouse that is 200 feet away. He looks at the top of the tall lighthouse with a 3 degree angle of elevation, but looks at the bottom of the tall lighthouse with 6 degree angle of depression. Find the height of the tall lighthouse.
draw a diagram
review the tan(x) function
You will see that the tall lighthouse has height
200 tan3° + 200 tan6°
??? I suspect a typo somewhere.
To find the height of the tall lighthouse, we can use the principles of trigonometry. Let's break down the problem into two parts:
1. Angle of elevation:
John is looking at the top of the tall lighthouse with a 3-degree angle of elevation. To calculate the height, we can use the tangent function. The tangent of an angle is the ratio of the opposite side to the adjacent side.
Let's assume the height of the tall lighthouse is h. We can set up the following equation:
tan(3 degrees) = h / 200
Solving for h, we have:
h = 200 * tan(3 degrees)
2. Angle of depression:
John is looking at the bottom of the tall lighthouse with a 6-degree angle of depression. Since the height is measured from the bottom of the lighthouse, we need to add the height of the lighthouse to the calculated value.
Let's assume the total height of the tall lighthouse is H. We can set up the following equation:
tan(6 degrees) = (H + h) / 200
Solving for H, we have:
H = 200 * tan(6 degrees) - h
Now, let's calculate the height of the tall lighthouse using the given angles:
h = 200 * tan(3 degrees)
h ≈ 200 * 0.0524
h ≈ 10.48 feet
H = 200 * tan(6 degrees) - h
H ≈ 200 * 0.1045 - 10.48
H ≈ 20.9 - 10.48
H ≈ 10.42 feet
Therefore, the height of the tall lighthouse is approximately 10.42 feet.