Chapter 4.3, Problem 67E

To determine

Find the total height of the building and distance of the viewer from the 86th floor.

Answer to Problem 67E

The total height of the building is $123+320.19=443.19$ meters

The distance of the viewer from the 86th floor is 323.33 meters

Explanation of Solution

Given:

A viewer is standing 45 meters away from a building. The angle of elevation to the top of the 86th floor is $\theta =82°.$ The total height of the building is 123 meters above the 86th floor.

To find the height of the building and the distance of the viewer from the 86th floor.

Let x be the height of the building up to 86th floor.

Let y be the distance of viewer from the 86th floor.

Now, x is the length of the side opposite to angle $\theta =82°.$

Length of the side adjacent to $\theta =82°.$ is 45 meters.

Now,

  $\tan \theta =\frac{\text{opposite side to}\theta }{\text{adjacent side to}\theta }$

Thus, substituting all the above values in this formula,

  $\tan 82°=\frac{x}{45}$

Or

  $x=45\times \tan 82°$

Now,

  $tan82°=7.1153$

Hence,

  $\begin{array}{l}x=45\times 7.1153\hfill \\ =320.19\hfill \end{array}$

Thus, the total height of the building is $123+320.19=443.19$ meters.

Now, y is the hypotenuse of the above given triangle.

  $\cos \theta =\frac{\text{adjacent side to}\theta }{\text{hypotenuse}}$

Thus, substituting all the above values in this formula,

  $\cos {82}^{\circ }=\frac{45}{y}$

Or,

  $\begin{array}{l}y=\frac{45}{\cos {82}^{\circ }}\hfill \\ \text{Now,}\cos {82}^{\circ }=0.139\hfill \\ \begin{array}{cc}\text{Hence,}y& =\frac{45}{0.139}\\ & =323.33\end{array}\hfill \end{array}$

Thus, distance of the viewer from the 86th floor is 323.33 meters.