Two trains start heading toward each other from two cities, the distance between which is 720 km, and meet right in the middle. The second train left 1 hour after the first train, but traveled at a speed 4 km/hour faster than the first train. Find the speed of both trains.
To solve this problem, we can use the concept of relative speed.
Let's assume the speed of the first train is x km/h. Since the second train is traveling 4 km/h faster, its speed would be x + 4 km/h.
The time taken by the first train to reach the meeting point can be calculated using the formula: time = distance / speed. In this case, the distance traveled by the first train is half of the total distance, which is 720/2 = 360 km. Therefore, the time taken by the first train is 360 / x.
The second train left 1 hour after the first train, so it traveled for 1 hour less. Therefore, the time taken by the second train is 360 / (x + 4) - 1.
Since both trains meet at the same time, their respective times should be equal. Therefore, we can set up the equation:
360 / x = 360 / (x + 4) - 1
Now, let's solve this equation to find the value of x:
Multiply through by x(x + 4) to eliminate the denominators:
360(x + 4) = 360x - x(x + 4)
Distribute and simplify:
360x + 1440 = 360x - x^2 - 4x
Rearrange the equation:
x^2 + 4x - 1440 = 0
Now, we have a quadratic equation. We can solve this equation by factoring, completing the square, or using the quadratic formula. After solving, we find that x ≈ 35.49 or x ≈ -39.49.
Since speed cannot be negative, we discard the negative value. Therefore, the speed of the first train is approximately 35.49 km/h.
To find the speed of the second train, we can substitute this value back into our initial assumption: x + 4 km/h. Thus, the speed of the second train is approximately 35.49 + 4 = 39.49 km/h.
Therefore, the speed of the first train is approximately 35.49 km/h, and the speed of the second train is approximately 39.49 km/h.
each went 720/2 = 360 km
train 1 time t
train 2 time ( t-2)
train 1 speed v
train 2 speed v+4
360 = v t so t = 360/v
360 = (v+4)(t-2) = vt -2v+4t-8
360 = v(360/v) - 2 v +4 (360/v) - 8
360 = 360 - 2 v + 1440/v - 8
2 v + 8 = 1440/v
2 v^2 + 8 v -1440 = 0
v^2 + 4 v - 720 = 0
v = -29 or + 25
one goes 25 the other 25+4 = 29
===================================
check
t = 360/25 = 14.4 hr
29*12.4 = 360 close enough