Two cars run at constant speeds around a one-mile racetrack. The faster car passes the slower car every two minutes if the cars run in the same direction. If they run in opposite directions, they meet every 15 seconds. Find the speed of the faster car in miles per hour.
To solve this problem, we can set up a system of equations.
Let's assume the slower car's speed is S (in miles per minute) and the faster car's speed is F (in miles per minute).
When the cars run in the same direction, the faster car passes the slower car every two minutes. This means that in two minutes, the faster car covers one lap more than the slower car. Since the racetrack is one mile long, the faster car covers one mile more than the slower car in two minutes. Therefore, the equation becomes:
2F - 2S = 1 ... Equation (1)
When the cars run in opposite directions, they meet every 15 seconds. In 15 seconds, the faster car covers a distance equal to its speed, and the slower car covers a distance equal to its speed. Since the racetrack is one mile long, the equation becomes:
15F + 15S = 1 ... Equation (2)
We have a system of two equations with two variables (F and S). We can solve this system to find the values of F and S.
First, let's isolate one variable from one of the equations. We'll solve Equation (1) for S:
2F - 2S = 1
-2S = 1 - 2F
S = (1 - 2F)/(-2)
S = (2F - 1)/2 ... Equation (3)
Now, substitute Equation (3) into Equation (2) to solve for F:
15F + 15[(2F - 1)/2] = 1
15F + 15F - 15/2 = 1
30F - 15/2 = 1
30F = (1 + 15/2)
30F = 17/2
F = (17/2) / 30
F = 17/60
Therefore, the speed of the faster car (F) is 17/60 miles per minute. To convert this to miles per hour, we multiply by 60 (since there are 60 minutes in an hour):
Speed of the faster car = (17/60) × 60
Speed of the faster car = 17 miles per hour
So, the speed of the faster car is 17 miles per hour.