12) Two jeeps in the desert leave their starting points at noon. The first travels a course of 345º and averages 23 mph. The second travels a course of 45º and averages 20 mph. How far apart are the two jeeps at 10 PM? What is the bearing from the first jeep to the second at this time?
jeep A ends up at (-230 sin15º,230 cos15º) = (-59.53,222.16)
jeep B ends up at (200 sin45º,200cos45º) = (141.42,141.42)
So, the distance between them is
√((141.42+59.53)^2 + (141.42-222.16)^2) = 216.56
The angle from A to B is θ where
tanθ = (141.42-222.16)/(141.42+59.53) = -0.40
As a compass bearing, that would be 112º
To find the distance between the two jeeps at 10 PM, we need to calculate the distance each jeep has traveled from noon to 10 PM.
First, let's calculate the distance traveled by the first jeep. The average speed of the first jeep is given as 23 mph. From noon to 10 PM, there are 10 hours. Therefore, the distance traveled by the first jeep is:
Distance = Speed × Time = 23 mph × 10 hours = 230 miles
Next, let's calculate the distance traveled by the second jeep. The average speed of the second jeep is given as 20 mph. Similarly, the distance traveled by the second jeep is:
Distance = Speed × Time = 20 mph × 10 hours = 200 miles
Now that we have the distances traveled by each jeep, we can use the concept of vectors to find the distance between them. We can treat the distance traveled by each jeep as the displacement vectors from their starting points at noon.
To find the distance between the two jeeps, we can use the Pythagorean theorem. The distance between the two jeeps is the magnitude of the resultant of the two displacement vectors:
Distance between jeeps = √(distance of Jeep 1)^2 + (distance of Jeep 2)^2
= √(230 miles)^2 + (200 miles)^2
= √(52900 miles^2 + 40000 miles^2)
= √(92900 miles^2)
≈ 305 miles
Therefore, the two jeeps are approximately 305 miles apart at 10 PM.
To find the bearing from the first jeep (Jeep 1) to the second jeep (Jeep 2) at this time, we need to determine the angular direction between them.
Since the bearing of Jeep 1 is 345º and the bearing of Jeep 2 is 45º, we can find the bearing from Jeep 1 to Jeep 2 by subtracting 45º from 345º:
Bearing from Jeep 1 to Jeep 2 = 345º - 45º
= 300º
Therefore, the bearing from the first jeep to the second jeep at 10 PM is 300º.