Exercise 3 Page 583

We are given that Annabelle and Rich are setting up decorations for their school dance, and we know that Rich is standing $5$ feet directly in front of Annabelle under a disco ball. Let $y$ represent the horizontal distance between Rich and the disco ball.

We know that the angle of elevation from Annabelle to the ball is $40^{\circ }$ and from Rich to the ball is $50^{\circ }.$ Since we are asked to evaluate how high the disco ball is, let $x$ represent this height. Notice that we have two right triangles in our diagram.

To evaluate the lengths of the missing sides, we can use one of the trigonometric ratios. Let's recall that the tangent of $\angle A$ is the ratio of the leg opposite $\angle A$ to the leg adjacent $\angle A.$ Using this definition, we can create the equations for $\tan 50^{\circ }$ and $\tan 40^{\circ }.$ As we can see, we need to solve the system of equations. To do this, we can use the Substitution Method. Our first step will be to isolate $x$ in the first equation. Next we will substitute the value of $x$ into the second equation. The value of $y$ is approximately $11.9$ feet. By substituting this value into the first equation, we can find the value of $x.$ The disco ball is approximately $14.2$ feet above the ground.