Chapter 8.2, Problem 47AYU

To determine

To find: The angle $\angle RPQ$ of the figure and the perpendicular distance from $R$ to $PQ$ .

Answer to Problem 47AYU

The value of $84.7°$ and the perpendicular distance from $R$ to $Q$ is $183.72\text{ft}$ .

Explanation of Solution

Given:

The height of the leaning tower of Pisa is $184.5$ feet.

The distance of the tower is $123$ feet from the base of the tower.

The angle of elevation of the top of the tower is $60°$ .

The given diagram is shown in Figure 1

Figure 1

Calculation:

Consider the diagram is shown in Figure 2

Figure 2

Consider the law of sine is,

  $\frac{\sin PQR}{PR}=\frac{\sin QRP}{PQ}$

Then,

  $\begin{array}{l}\frac{\sin 60°}{184.5}=\frac{\sin QRP}{123}\\ \sin QRP=\left(0.004694\right)\left(123\right)\\ \sin QRP=0.5774\\ \angle QRP=35.265°\end{array}$

Consider the sum angle property is,

  $\angle PQR+\angle QRP+\angle RPQ=180°$

Then,

  $\begin{array}{l}60°+35.265°+\angle RPQ=180°\\ \angle RPQ=84.7°\end{array}$

From the figure the value of $h$ is,

  $\begin{array}{l}\sin 84.7°=\frac{h}{184.5}\\ h=183.72\end{array}$

Thus, the value of $84.7°$ and the perpendicular distance from $R$ to $Q$ is $183.72\text{ft}$ .