Chapter 4.8, Problem 29E

To determine

Find the altitude of plane.

Answer to Problem 29E

  $\boxed{3.32ft}$

Explanation of Solution

Given information:

You observe a plane approaching overhead and assume that its speed is $\text{55}0$ miles per hour. The angle of elevation of the plane is $16°$ at one time and $57°$ one minute later. Approximate the altitude of the plane.

Calculation:

The figure can be drawn as a right angled triangle as shown below.

In the above figure $AB=DE=h$ is the altitude at which the plane is flying and the angles of depression from $A$ and $D$ are $16°$ and $54°$ resp.

Therefore angles of elevation from $C$ to $A$ and $D$ are $16°$ and $54°$ resp. Let $BC=x$

We will use the fact that the speed of the plane is

  $550miles/hr=\frac{550}{60}miles/\min$

Therefore, $BE=1\times \frac{55}{6}miles$

  $=\frac{55}{6}miles$

In the right $\Delta ABC$ , we shall use the fact that $\theta =16°$ .

  $\tan 16=\frac{AB}{BC}$

  $BC=\frac{AB}{\tan 16}\mathrm{.....}\left(1\right)$

  $x=\frac{AB}{\tan 16}$

For the right $\Delta DEC$

  $\tan 54=\frac{DE}{EC}$

  $\tan 54=\frac{DE}{EC}$

  $x-\frac{55}{6}=\frac{h}{\tan 54}$

From (1) and (2) we get

  $\frac{h}{\tan 16}=\frac{h}{\tan 54}-\frac{55}{6}$

  $\Rightarrow h\approx 3.32$

Hence, the altitude of the plane is $\boxed{3.32ft}$