Two hiking paths diverge at an angle of 75°. If one person took a path and walked 5 km/h and another person took the other path and walked at 6 km/h, how far apart are they after 1½ hours?
Draw a triangle with your one angle as 75 degrees and the two adjacent sides as how far each person walked after 1.5 hrs. Then law of cosines
c^2=7.5^2+9^2- 2(9)(7.5)cos75
solve for c
use law of cosines. Your distance z can be found by
z/1.5 = 5^2 + 6^2 - 2*5*6 cos75°
D1 = V1 * t = 5km/h[75o] * 1.5h = 7.5 km[75o].
D2 = V2 * t = 6km/h[0o] * 1.5h = 9 km[0o].
D = D2 - D1 = 9[0o] - 7.5[75o] = 9 - 1.94 - 7.24i = 7.06 - 7.24i.
D = sqrt(X^2+Y^2) = sqrt(7.06^2+7.24^2) =
To find out how far apart the two people are after 1½ hours, we need to calculate the distances each person has traveled.
First, let's determine how far each person has traveled in 1½ hours.
Person 1, who is walking at a speed of 5 km/h, will have traveled:
Distance = Speed × Time
Distance = 5 km/h × 1.5 hours
Distance = 7.5 km
Person 2, who is walking at a speed of 6 km/h, will have traveled:
Distance = Speed × Time
Distance = 6 km/h × 1.5 hours
Distance = 9 km
Now that we know the distances traveled by each person, we can figure out how far apart they are.
To do this, we can use the concept of vector addition. We can represent the paths taken by the two people as vectors with magnitudes equal to the distances traveled and directions given by the angle of divergence.
Let's denote person 1's path as vector P1 and person 2's path as vector P2.
From the information given, we know that the two vectors diverge at an angle of 75°. Therefore, the angle between P1 and P2 is also 75°.
To find the distance between the two people, we can use the law of cosines. According to this law:
Distance^2 = P1^2 + P2^2 - 2(P1)(P2)cosθ
Where P1 and P2 are the magnitudes of the vectors, and θ is the angle between them.
Plugging in the values we have:
Distance^2 = (7.5 km)^2 + (9 km)^2 - 2(7.5 km)(9 km)cos(75°)
Now we can solve for the distance:
Distance^2 = 56.25 km^2 + 81 km^2 - 135 km^2(cos75°)
Distance^2 = 137.25 km^2 - 135 km^2(cos75°)
Distance^2 = 137.25 km^2 - 135 km^2(0.2588190451)
Distance^2 ≈ 137.25 km^2 - 35.0807 km^2
Distance^2 ≈ 102.1693 km^2
Distance ≈ √102.1693 km
Distance ≈ 10.108 km
After 1½ hours, the two people are approximately 10.108 km apart.