A person in a lighthouse notices two boats out on the ocean. The angle of depression to the further boat is 20 degrees while the angle of depression to the closer boat is 75 degrees. What is the distance between the two boats if the observer in the lighthouse is 60 feet off the ground?
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To solve this problem, we need to use the concept of trigonometry and specifically the tangent function. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Let's denote the distance between the observer in the lighthouse and the further boat as x feet, and the distance between the observer and the closer boat as y feet.
From the given information, we know that the angle of depression to the further boat is 20 degrees, and the angle of depression to the closer boat is 75 degrees. We also know that the observer in the lighthouse is 60 feet off the ground.
Now, let's look at the triangle formed by the observer, the further boat, and the ground. The vertical side of the triangle represents the height of the observer (60 feet), and the horizontal side represents the distance between the observer and the further boat (x feet). The angle between these two sides is the angle of depression to the further boat (20 degrees).
Using the tangent function, we can set up the following equation:
tan(20 degrees) = opposite/adjacent = 60/x
Now, let's look at the triangle formed by the observer, the closer boat, and the ground. Similar to before, the vertical side of the triangle represents the height of the observer (60 feet), and the horizontal side represents the distance between the observer and the closer boat (y feet). The angle between these two sides is the angle of depression to the closer boat (75 degrees).
Using the tangent function again, we can set up the following equation:
tan(75 degrees) = opposite/adjacent = 60/y
Now we have a system of two equations:
1) tan(20 degrees) = 60/x
2) tan(75 degrees) = 60/y
To solve this system of equations, we can rearrange equation (1) to find x in terms of y:
x = 60/tan(20 degrees)
Plug this value of x into equation (2) and solve for y:
tan(75 degrees) = 60/(60/tan(20 degrees))
Now, using a calculator or a trigonometric table, find the values of tan(20 degrees) and tan(75 degrees).
Once you have those values, substitute them into the equation and calculate the value of y:
tan(75 degrees) ≈ 3.7321
y = 60/(60/3.7321) ≈ 3.7321
Therefore, the distance between the two boats is approximately 3.7321 feet.
Please note that due to rounding errors, the final answer may have slight discrepancies.