Final answer:
The height of the lifeguard station can be estimated using trigonometry, specifically the tangent function with the angles of elevation and distances given, by setting up two equations for the same height and solving them.
Step-by-step explanation:
To estimate the height of the lifeguard station, we can use trigonometric functions, specifically the tangent function, which relates the angle of elevation to the opposite side and adjacent side in a right-angled triangle. The problem states that one person is 15 feet closer to the station than you, creating two right-angled triangles with a common height, which is the height of the lifeguard station.
Let's denote the height of the lifeguard station as h, the horizontal distance from you (the further person) to the station as d, and the horizontal distance from the other person to the station as d - 15 feet. Using the angles of elevation and the definition of the tangent function:
- For you: tan(28°) = h / d
- For the other person: tan(52°) = h / (d - 15)
We now have two equations with two unknowns. To solve for the height h, we need to express d in terms of h using the first equation and substitute into the second one:
d = h / tan(28°)
Substituting d into the second equation gives:
tan(52°) = h / (h / tan(28°) - 15)
By solving this equation for h, we can find the height of the lifeguard station. We can solve this equation using algebraic manipulation and trigonometric identities to find the precise value of h.