Chapter 6.1, Problem 48E

To determine

To find:The distance from the gazebo to the dock.

Answer to Problem 48E

The distance from the Gazebo to the dock is $72.47m$ . $42.43km$ .

Explanation of Solution

Given information:

A bridge is built across a small lake from a gazebo to a dock. The bearing from the gazebo to the dock is $\text{S}41°\text{W}$ . From a tree 100 meters from gazebo, the bearings to the gazebo and the dock are $\text{S}74°\text{E}$ and $\text{S28}°\text{E}$ , respectively.

  

Calculations:

Drawing the given scenario,

  

Here, T is Tree, G is Gazebo and D is Dock.

Thus, in $\Delta TGD$ ,

  $\begin{array}{l}\angle T=74-28=46°\\ \angle G=180-74-28-41=37°\end{array}$

Using this,

  $\begin{array}{l}\angle D=180-\angle T-\angle G(\because \text{Angle sum property})\\ \Rightarrow \angle D=180-46-37\\ \Rightarrow \angle D=97°\end{array}$

Using law of sines

  $\begin{array}{l}\frac{\sin (T)}{DG}=\frac{\sin (G)}{TD}=\frac{\sin (D)}{TG}\\ \Rightarrow \frac{\sin (46)}{DG}=\frac{\sin (36)}{TD}=\frac{\sin (97)}{100}\end{array}$

To find $DG$ :

  $\begin{array}{l}\frac{\sin (46)}{DG}=\frac{\sin (97)}{100}\\ \Rightarrow DG=\frac{100\sin (46)}{\sin (97)}\\ \Rightarrow DG=72.47\end{array}$

Conclusion:

Therefore, the distance from the Gazebo to the dock is $72.47m$ .