Two cars run at constant speeds around a one-mile racetrack. If the cars circle the track in the same direction, the faster car passes the slower car every 10 minutes. If the cars circle the track in opposite directions, the cars meet every 30 seconds. Find the speed of the faster car in miles per hour.
Note:
The racetrack is 'circular' in the sense that the start and end points are the same, and the cars are running laps around the racetrack at a constant speed.
If the track has circumference c miles, then and the car's speeds are x and y mi/s, then since time = distance/speed,
In the same direction, the slow car has a "lead" of 1 mile which the faster car has to make up.
1/(y-x) = 120
1/(x+y) = 30
120(y-x) = 30(x+y)
120y-120x = 30x+30y
90y = 150x
y = 5/3 x
1/(x + 5/3 x) = 30
8/3 x * 30 = 1
80x = 1
x = 1/80 mi/sec = 45 mph
y = 75 mph
To find the speed of the faster car, we can use the concept of relative speed and the formula for finding the distance covered in a given time.
Let's call the speed of the slower car S miles per hour and the speed of the faster car F miles per hour.
When the cars are moving in the same direction, the faster car passes the slower car every 10 minutes. This means that in 10 minutes, the faster car covers a distance equal to the length of the racetrack (1 mile) plus the distance covered by the slower car. The distance covered by the slower car in 10 minutes is equal to its speed times the time, which is S * 10/60.
Therefore, when the cars are moving in the same direction, the distance covered by the faster car in 10 minutes is 1 + S * 10/60 miles.
When the cars are moving in opposite directions, they meet every 30 seconds. This means that in 30 seconds, the cars together cover a distance equal to the length of the racetrack (1 mile). Since the faster car covers the entire distance in that time, its speed is 1 mile per 30 seconds.
Now, we can set up two equations based on the information:
Equation 1: F * 10/60 = 1 + S * 10/60 (distance covered by the faster car in 10 minutes)
Equation 2: F * (30/60) = 1 (distance covered by the faster car in 30 seconds)
Let's solve these equations to find the value of F (speed of the faster car).
From Equation 1, we can simplify:
F * 10/60 - S * 10/60 = 1
F * (10/60 - S/60) = 1
F * (10 - S)/60 = 1
F = 60/(10 - S)
Now, let's substitute this value of F into Equation 2:
60/(10 - S) * (30/60) = 1
30/(10 - S) = 1
10 - S = 30
S = 10 - 30
S = -20
Since the speed of a car cannot be negative, this value of S is not valid. Therefore, it means that the cars cannot be moving in opposite directions.
Hence, the speed of the faster car in miles per hour is given by the formula:
F = 60/(10 - S)
Substituting S = 10 into this formula:
F = 60/(10 - 10)
F = 60/0 (division by zero is undefined)
Therefore, we cannot determine the speed of the faster car using the given information.