The treasure map in the figure gives the following directions to the buried treasure: "Start at the old oak tree, walk due north for 490 paces, then due east for 120 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 ^\circ east of north. After walking 330 paces you see an opening through the woods. Which direction should you go to reach the treasure? How far should you go to reach the treasure?

To determine the direction and distance to reach the treasure, we need to use vector addition. Here's how you can find the solution:

1. Draw a diagram: Start by drawing a diagram that represents the given situation. Mark the old oak tree, the dragon, the yellow brick road, and the opening through the woods. Label the distances and angles as mentioned in the problem.

2. Find the displacement: Calculate the displacement between the starting point (old oak tree) and where you are after walking 330 paces along the yellow brick road. To do this, break down the displacement into its northward and eastward components. Since you are walking at an angle of 60 degrees east of north, the northward component would be 330 * sin(60°), and the eastward component would be 330 * cos(60°).

3. Account for the original directions: Now, you need to account for the original directions mentioned in the treasure map. Start by subtracting the original northward displacement of 490 paces (as stated in the map) from the northern displacement calculated in step 2. This will give you the remaining northward displacement. Similarly, subtract the original eastward displacement of 120 paces from the eastward displacement calculated in step 2. This will give you the remaining eastward displacement.

4. Determine the angle and distance: Use inverse trigonometric functions to find the angle between the remaining northward displacement and the remaining eastward displacement. The angle can be found by taking the inverse tangent of the quotient of the remaining northward displacement divided by the remaining eastward displacement. This angle will represent the direction you should go to reach the treasure.

5. To find the distance, use the Pythagorean theorem: The remaining northward displacement and the remaining eastward displacement form the legs of a right-angled triangle. Use the Pythagorean theorem to find the hypotenuse, which represents the distance you need to travel to reach the treasure.

By following these steps, you should be able to determine the direction and distance needed to reach the treasure while avoiding the dragon.

To determine the direction and distance to reach the treasure, let's break down the given information and analyze the situation step-by-step.

Step 1: Start at the old oak tree and walk due north for 490 paces.
Step 2: Instead of walking due north, you choose to walk along the yellow brick road at an angle 60 degrees east of north.
Step 3: After walking 330 paces, you see an opening through the woods.

To find the direction to the treasure:

Step 1: Determine the angle between due north and the yellow brick road.
- The angle between due north and the yellow brick road is 60 degrees.

Step 2: Add the angle of 60 degrees to the original direction of due east.
- This means the angle between due east and the yellow brick road is (90 degrees + 60 degrees) = 150 degrees.

Therefore, you should go in the direction of 150 degrees from the yellow brick road to reach the treasure.

To find the distance to the treasure:

Step 1: Calculate the horizontal and vertical components of the distance you've traveled along the yellow brick road.
- The horizontal component is 330 paces * cos(60 degrees) = 165 paces.
- The vertical component is 330 paces * sin(60 degrees) = 285.27 paces (rounded to two decimal places).

Step 2: Determine the remaining horizontal and vertical distance to the treasure.
- The remaining horizontal distance is 120 paces - 165 paces = -45 paces (negative because it lies in the opposite direction).
- The remaining vertical distance is 490 paces - 285.27 paces = 204.73 paces (rounded to two decimal places).

Step 3: Use Pythagoras' theorem to find the total distance to the treasure.
- The total distance is sqrt((-45 paces)^2 + (204.73 paces)^2) = 219.14 paces (rounded to two decimal places).

Therefore, to reach the treasure, you should travel in the direction of 150 degrees, and the total distance to the treasure is approximately 219.14 paces.

Nevermind! I got it!