The treasure map 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 east of north. After walking 370 paces you see an opening through the woods.
(1)Which direction should you go to reach the treasure? (answer in degrees west of north)
(2)How far should you go to reach the treasure? (answer in paces)
To find the answers, we need to break down the information given and analyze the situation step by step:
1. Start at the old oak tree and walk due north for 490 paces.
2. Then walk due east for 120 paces.
3. At this point, we encounter an angry dragon just north of the tree and decide to avoid it by following the yellow brick road.
4. We set off at an angle east of north and walk 370 paces.
5. After 370 paces, we see an opening through the woods.
Now let's find the answers to the questions:
(1) Which direction should you go to reach the treasure? (answer in degrees west of north)
To determine this, we need to consider the path we have walked so far. We initially walked due north for 490 paces, then due east for 120 paces. This forms a right-angled triangle. The angle we walked along the yellow brick road can be found using trigonometry.
Using the tangent function, we can calculate the angle:
tan(angle) = (opposite side) / (adjacent side)
tan(angle) = (370 paces) / (490 paces)
angle = arctan(370/490) ≈ 38.33 degrees
Therefore, the direction we need to go is 38.33 degrees west of north.
(2) How far should you go to reach the treasure? (answer in paces)
To determine the distance, we can use trigonometry as well. We have already determined the angle formed between the yellow brick road and the north direction.
We can use the sine function to determine the distance:
sin(angle) = (opposite side) / (hypotenuse)
sin(angle) = (unknown distance) / (distance walked from the old oak tree to the current point)
sin(38.33 degrees) = (unknown distance) / (490 paces)
unknown distance = sin(38.33 degrees) * 490 paces ≈ 299.13 paces
Therefore, to reach the treasure, you should go approximately 299.13 paces.