Chapter 8.2, Problem 41AYU

To determine

Find the height of the airplane.

Answer to Problem 41AYU

  $\boxed{381.6ft}$

Explanation of Solution

Given information:

An aircraft is spotted by two observers who are $1000ft$ apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane, as indicated in figure. How high is the airplane?

  

Calculation:

Consider the figure, and sum up the all angles of triangle,

Angle A is,

  $\begin{array}{l}40°+35°+A=180°\\ A=180°-75°\\ A=105°\end{array}$

Apply law of sine to find side of triangle,

  $\begin{array}{l}\frac{\sin A}{a}=\frac{\sin B}{b}\\ \frac{\sin 105°}{1000}=\frac{\sin 35°}{b}\\ 0.0009659=\frac{\sin 35°}{b}\\ b\left(0.0009659\right)=\sin 35°\\ b=\frac{\sin 35°}{0.0009659}\\ b=593.8\end{array}$

Now height of the plane is from the ground is,

  $h=b\sin 40°$ $b$ is the one side of triangle.

Substitute $b=593.8$ we get,

  $\begin{array}{l}h=593.8\sin 40°\\ h=381.6ft\end{array}$

Hence the height of the plane from the ground is $\boxed{381.6ft}$