A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 13°32'. When the boat stops, the angle of depression is 45°10'. The lighthouse is 160 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.
Draw a diagram. You will see that if the distance moved is x, then
x = 160cot13°32' - 160cot45°10'
To find the distance the boat traveled, we need to use trigonometric ratios. We will use the tangent function, which is defined as the opposite side divided by the adjacent side in a right triangle.
Let's label the height of the lighthouse as "h" and the distance the boat traveled as "d". We have two right triangles, one at the beginning and one at the end:
1. At the beginning:
- Angle of depression = 13°32'
- Opposite side = h (height of the lighthouse)
- Adjacent side = d (distance traveled by the boat)
- Tan(13°32') = h / d
2. At the end:
- Angle of depression = 45°10'
- Opposite side = h (height of the lighthouse)
- Adjacent side = d (distance traveled by the boat)
- Tan(45°10') = h / d
We can solve these two equations simultaneously to find the value of "d". Here's how:
1. Convert the angles from degrees to decimal form:
- Angle of depression at the beginning = 13°32' = 13 + 32/60 = 13.5333°
- Angle of depression at the end = 45°10' = 45 + 10/60 = 45.1667°
2. Substitute the values into the equations:
- Tan(13.5333°) = 160 / d
- Tan(45.1667°) = 160 / d
3. Solve for "d" by isolating it:
- Rearrange the equations:
- d = 160 / Tan(13.5333°)
- d = 160 / Tan(45.1667°)
4. Use a calculator to evaluate the trigonometric functions:
- d ≈ 1108.29 feet (rounded to two decimal places)
Therefore, the boat traveled approximately 1108.29 feet from when it was first noticed until it stopped.