Silvio is an air-traffic controller who must calculate the altitude of an incoming plane. He sees the plane at an angle of elevation of 40 as it flies over a signal tower. Silvio knows the signal tower is 3 miles away, so he creates a trigonometric model to calculate the altitude of the plane relative to his own position. Part A: Did Silvio choose the most appropriate function family for his model? Part B: Which function family correctly models this situation?

Since the problem is not telling us the height of Silvio, we are going to assume it is not relevant for our calculations.
Let [tex]h[/tex] the altitude of the incoming plane. We know for our problem that the distance between Silvio and the tower is 3 miles, Also we know that the angle of elevation to the plane is 40°. With this information we can create a triangle as shown in the figure. We need a function that relates the angle of elevation with its opposite and adjacent sides, that function is tangent.
[tex]tan \alpha = \frac{opposite.side}{adjacent.side} [/tex]
[tex]tan(40)= \frac{h}{3} [/tex]
[tex]h=3tan(40)[/tex]
[tex]h=2.5miles[/tex]

We can conclude that we should use the trig function tangent to model this situation; also, we can conclude that the equation that describes this situation is [tex]h=3tan(40)[/tex].