Question

The treasure map in FIGURE P3.40 gives the following directions to the buried treasure: "Start at the old oak tree, walk due north for 500 paces, then due east for 100 paces. Dig." But when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle ${60}^{°}$east of north. After walking 300 paces you see an opening through the woods. Which direction should you go, and how far, to reach the treasure?

Short answer

The treasure first, move 300 paces on the yellow road, deviate in a direction of ${30}^{°}$west of north and move 412.4 paces.

Step-by-step solution

Step 1. Given information

The distance to be travelled to get the treasure is BT and is as shown in the figure.

Treasure distance is 500 paces north and 100 paces east.

In any quadrangle sum of opposite angles is${180}^{°}$ , and Pythagoras theorem.

Step 2. Explanation

From the figure$\angle \text{OAT}={90}^{°},\angle \text{ATB}={120}^{°},\angle \text{TBO}={90}^{°}$ and$\angle \text{BOA}={60}^{°}$

From the east, a yellow brick road is inclined by ${30}^{°}$. So, to get the treasure first we have to move 300 paces on the yellow road, deviate in a direction ${30}^{°}$of west of north and travel a distance of BT (to be calculated). The calculation of BT is shown below.

From figure using Pythagoras theorem,

$\begin{array}{l}\text{OT}=\sqrt{{\text{OA}}^2+{\text{AT}}^2}\\ =\sqrt{{500}^2+{100}^2}\\ =509.9\simeq 510\text{paces.}\\ \\ \text{OT}=\sqrt{{\text{OB}}^2+{\text{TB}}^2}\text{TB}\\ =\sqrt{{\text{OT}}^2-{\text{OB}}^2}\\ =\sqrt{{510}^2-{300}^2}\\ =412.4\text{paces.}\end{array}$

Conclusion:

To get the treasure first, move 300 paces on the yellow road, deviate in a direction of ${30}^{°}$west of north and move 412.4 paces.