An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for 5.7 km, but when the snow clears, he discovers that he actually traveled 8.4 km at 55o north of due east. How far (in km) must he now travel to reach base camp?
He has traveled 8.4 sin55 = 6.88 km north and
8.4 cos55 = 4.82 km east.
The distance he must travel to base camp is 6.88-5.7 = 1.18 km south and 4.82 km west.
Take the square root of the sum of the squares for the distance he has to travel.
That would be 4.96 km
To find out how far the explorer must now travel to reach base camp, we can use trigonometry and the given information.
1. We know that the explorer traveled 8.4 km at 55° north of due east.
2. Since the explorer was supposed to travel due north for 5.7 km, we can calculate the eastward distance that he veered off track: distance veered off track = 5.7 km * sin(55°).
This gives us: distance veered off track ≈ 5.7 km * 0.8191 ≈ 4.6691 km.
3. To find the straight-line distance between his ending point and the base camp, we can use the Pythagorean theorem.
Let's call this distance "d."
d^2 = (5.7 km)^2 + (distance veered off track)^2.
Simplifying the equation, we have: d^2 = 32.49 km^2 + 21.7606 km^2 ≈ 54.2506 km^2.
4. Taking the square root of both sides of the equation, we find:
d ≈ √(54.2506 km^2).
d ≈ 7.366 km.
Therefore, the explorer must now travel approximately 7.366 km to reach the base camp.
To solve this problem, we can use trigonometry and basic geometry concepts. Let's break it down step by step:
1. Start by visualizing the explorer's path. He was supposed to travel due north for 5.7 km. Let's mark this distance on a diagram:
N
|
|
| 5.7 km
-----
2. According to the problem, the explorer actually ended up traveling 8.4 km at an angle of 55° north of due east. Let's mark this path on the diagram as well:
N |
| \
| \
| 8.4 km \
---------------
3. Now, we need to determine the distance and direction in which the explorer must travel to reach the base camp from his current position (after the detour). To do this, we'll use trigonometry.
4. We can see that the angle between the original straight path and the detoured path is 35° (90° - 55°). Using this angle, we can split the 8.4 km into its north and east components.
north component = 8.4 km * sin(35°)
east component = 8.4 km * cos(35°)
5. Calculate these components:
north component = 8.4 km * sin(35°) = 4.791 km
east component = 8.4 km * cos(35°) = 6.963 km
6. Now, we can determine the remaining distance the explorer needs to travel to reach the base camp. Since the explorer needs to travel due north to reach the base camp, the remaining distance will be the north component we calculated earlier, which is 4.791 km.
Therefore, the explorer must travel approximately 4.791 km to reach the base camp.
It's worth noting that the answer might not be exact due to rounding errors during the calculations.