Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker to the landmark meter are 37° and 21°. How far apart are the hikers? Round your answer to the nearest whole meter. 1064 m 2064 m 2063 m 2065 m
The correct answer and explanation is :
To find how far apart the two hikers are, we can model this as a trigonometry problem involving right triangles.
Step 1: Visualize the problem
- Each hiker is on opposite sides of a canyon and 525 meters above the canyon floor.
- There’s a landmark directly on the canyon floor between them.
- The angles of depression from each hiker to the landmark are 37° and 21°.
This gives us two right triangles sharing a base (the horizontal ground between the two hikers), with one side at 525 meters high and angles of depression of 37° and 21° respectively.
Step 2: Apply trigonometry
Using right triangle trigonometry, particularly the tangent function:
(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}})
For each hiker:
Let:
- (x_1) = horizontal distance from the hiker at 37° to the landmark
- (x_2) = horizontal distance from the hiker at 21° to the landmark
Using the tangent definition:
- (\tan(37^\circ) = \frac{525}{x_1} \Rightarrow x_1 = \frac{525}{\tan(37^\circ)})
- (\tan(21^\circ) = \frac{525}{x_2} \Rightarrow x_2 = \frac{525}{\tan(21^\circ)})
Step 3: Compute distances
- (\tan(37^\circ) \approx 0.7536 \Rightarrow x_1 \approx \frac{525}{0.7536} \approx 696.97 \, \text{m})
- (\tan(21^\circ) \approx 0.3839 \Rightarrow x_2 \approx \frac{525}{0.3839} \approx 1367.68 \, \text{m})
Add them to find the total distance between hikers:
(x_1 + x_2 \approx 696.97 + 1367.68 \approx 2064.65)
Step 4: Final Answer
Rounded to the nearest whole meter, the distance between the hikers is 2065 meters.
✅ Correct Answer: 2065 m ✅
This trigonometric method works because it relies on basic right triangle relationships and the consistent elevation of both hikers. It’s a classic example of applying tangent in real-world navigation or surveying contexts.