Answer:
Let's call the distance between the lighthouse and the first boat "x" and the distance between the lighthouse and the second boat "y". We want to find the distance between the two boats, which we can call "d".
From the information given, we can draw the following diagram:
/|
/ |
/ |y
/ |
/θ2 |
/____|
L / x /B2
θ1
/|
/ |
/ |y-x
/ |
/θ3 |
/____|
A1 B1
Where L is the lighthouse, A1 and B1 are the positions of the first and second boats respectively, θ1 is the angle of depression to the first boat, θ2 is the angle of depression to the second boat, θ3 is the angle between the two boats, and x and y are as defined above.
From the diagram, we can see that:
tan(θ1) = 80 / x
tan(θ2) = 80 / y
tan(θ3) = (y - x) / 80
We also know that the two boats are in a straight line, so the sum of the distances to each boat is equal to the distance between the boats:
x + y = d
To solve for d, we need to eliminate x and y from these equations. We can start by solving for x and y in terms of θ1 and θ2:
x = 80 / tan(θ1)
y = 80 / tan(θ2)
Substituting these expressions into the equation for d, we get:
d = x + y
= 80 / tan(θ1) + 80 / tan(θ2)
Now we just need to plug in the values given in the problem:
θ1 = 55°
θ2 = 40°
d = 80 / tan(55°) + 80 / tan(40°)
≈ 196.8 feet
Therefore, the distance between the two boats is approximately 196.8 feet.