Leroy is 6.5 ft tall and standing on a bridge. He looks down with an angle of depression of 31 degrees towards a boat traveling east. He then turns around and looks down with an angle of depression of 24 degrees towards a boat traveling west. If the boats are 432 ft apart, how far away from Leroy is the second boat? How do I set up this problem and explain please? Thank you!
If Leroy's eyes are h feet above the water, then we can make the diagram as follows:
T is the top of Leroy's head, x feet above B, directly below him on the water.
E is the boat eastward
W is the boat westward.
Now we have
x/BW = tan 24°
x/BE = tan 31°
eliminating x, we have
BW tan 24° = BE tan 31°
But, BW+BE = 432, so
BW tan 24° = (432-BW)tan 31°
BW = 432tan31°/(tan24°+tan31°) = 248.14
The distance wanted is TW, given by
BW/TW = cos 24°
TW = BW/cos24° = 248.14/0.9135 = 271.62
To solve this problem, we can set up two right triangles. Let's start by setting up the first triangle when Leroy looks towards the boat traveling east.
In this triangle:
- Leroy's height is the opposite side.
- The distance from Leroy to the boat is the adjacent side.
- The angle between Leroy's line of sight and the horizontal line is the angle of depression.
We can use the tangent function to relate the angle of depression and the sides of the triangle. The tangent function is defined as the ratio of the opposite side to the adjacent side:
tan(angle) = opposite / adjacent
Applying this to the first triangle:
tan(31 degrees) = Leroy's height / distance to the boat (let's call this distance a)
Now, let's set up the second triangle when Leroy looks towards the boat traveling west:
In this triangle:
- Leroy's height is the opposite side.
- The distance from Leroy to the second boat is the adjacent side.
- The angle between Leroy's line of sight and the horizontal line is the angle of depression.
Similarly, applying the tangent function to the second triangle:
tan(24 degrees) = Leroy's height / distance to the second boat (let's call this distance b)
From the problem, we know that the distance between the two boats is 432 ft. We can use this information to set up an equation:
distance to the boat + distance to the second boat = 432
Using the variables we defined earlier:
a + b = 432
Now we have two unknowns (a and b) and two equations that relate them. We can solve these equations simultaneously to find the values of a and b.
Let's summarize the setup:
1. Set up the first triangle using the tangent function with the angle of depression of 31 degrees.
2. Set up the second triangle using the tangent function with the angle of depression of 24 degrees.
3. Set up the equation using the distance between the two boats, where a and b represent the distances to each boat.
4. Solve the equations simultaneously to find the values of a and b.
5. Once you have the values of a and b, you can determine how far away from Leroy the second boat is (distance b).