While camping, Henry dropped his keys. Henry walks 45 yards due east and then turns 25° north of east and walks an additional 30 yards. Aram walks 45 yards due west and then walks... While camping, Henry dropped his keys. Henry walks 45 yards due east and then turns 25° north of east and walks an additional 30 yards. Aram walks 45 yards due west and then walks 38° north of west another 30 yards. At this point, who is closer to the campsite?
Understand the Problem
The question is asking which person, Henry or Aram, is closer to the campsite after both have walked specific distances in different directions. To solve this, we will calculate the final positions of both individuals using their starting point and the specified angles and distances.
Answer
Henry is closer to the campsite.
Answer for screen readers
Henry is closer to the campsite.
Steps to Solve
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Calculate Henry's Final Position
Henry walks 45 yards due east. His coordinates after this step are: $$(x_H, y_H) = (45, 0)$$
Then he turns 25° north of east and walks an additional 30 yards. We can calculate the change in position using trigonometry:
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For the x-coordinate (east-west): $$ \Delta x_H = 30 \times \cos(25°) $$
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For the y-coordinate (north-south): $$ \Delta y_H = 30 \times \sin(25°) $$
Therefore, Henry's final coordinates are: $$(x_H, y_H) = (45 + \Delta x_H, 0 + \Delta y_H)$$ $$ \Delta x_H = 30 \times \cos(25°) $$ $$ \Delta y_H = 30 \times \sin(25°) $$
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Calculate Aram's Final Position
Aram walks 45 yards due west. His coordinates after this step are: $$(x_A, y_A) = (-45, 0)$$
He then turns 38° north of west and walks an additional 30 yards. Again, using trigonometry:
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For the x-coordinate (east-west): $$ \Delta x_A = 30 \times \cos(180° - 38°) = -30 \times \cos(38°) $$
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For the y-coordinate (north-south): $$ \Delta y_A = 30 \times \sin(180° - 38°) = 30 \times \sin(38°) $$
Therefore, Aram's final coordinates are: $$(x_A, y_A) = (-45 + \Delta x_A, 0 + \Delta y_A)$$ $$ \Delta x_A = -30 \times \cos(38°) $$ $$ \Delta y_A = 30 \times \sin(38°) $$
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Calculate Distance from Campsite
The distance from the campsite (the origin) for both Henry and Aram can be calculated using the distance formula:
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For Henry: $$ D_H = \sqrt{(x_H)^2 + (y_H)^2} $$
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For Aram: $$ D_A = \sqrt{(x_A)^2 + (y_A)^2} $$
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Compare Distances
Finally, compare $D_H$ and $D_A$ to determine who is closer to the campsite.
Henry is closer to the campsite.
More Information
This problem involves using trigonometric functions to determine the positions after moving in specified directions, then applying the distance formula to compare the final distances from a reference point, in this case, their starting campsite.
Tips
- Forgetting to convert angles to radians when using a calculator (if needed).
- Incorrectly applying signs for the coordinates when moving in negative directions.
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