Find the speed of a train if an increase of 6kph lowers the time for a journey of 72 km by 10 minutes.
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72/s = 72/(s+6) + 1/6
s^2 + 6s - 2592 = 0
(s+54)(s-48) = 0
speed = 48km/hr
time at 48km/hr: 1.5 hr = 90 min
time at 54km/hr: 4/3 hr = 80 min
To find the speed of the train, we need to apply the formula:
Speed = Distance/Time
Let's assume the original speed of the train is 'x' km/h.
Case 1: Original Journey
Distance = 72 km
Time = (Distance / Speed) = 72/x hours
Case 2: Increased Speed
Distance = 72 km
Time = (Distance / (Speed + 6)) = 72/(x + 6) hours
According to the problem, the difference in time between the original journey and the increased speed journey is 10 minutes or 10/60 = 1/6 hours.
So, the equation can be set up as:
72/x - 72/(x + 6) = 1/6
Now, let's solve the equation to find the value of x.
Multiplying every term by 6(x)(x+6) to eliminate the fractions:
6(x+6) - 6x = (x)(x+6)/6
6x + 36 - 6x = x^2 + 6x/6
Simplifying,
36 = x^2 + 6x
Rearranging,
x^2 + 6x - 36 = 0
To solve the quadratic equation, we can use factoring or the quadratic formula. Factoring does not result in nice whole numbers, so let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For the equation x^2 + 6x - 36 = 0,
a = 1, b = 6, c = -36
Plugging these values into the quadratic formula, we get:
x = (-6 ± sqrt(6^2 - 4(1)(-36))) / 2(1)
x = (-6 ± sqrt(36 + 144)) / 2
x = (-6 ± sqrt(180)) / 2
x = (-6 ± 12.727) / 2
Now, we have two potential values for x:
x = (-6 + 12.727) / 2 = 3.727
x = (-6 - 12.727) / 2 = -9.727
Since speed cannot be negative, the original speed of the train is 3.727 km/h.
Therefore, the speed of the train is approximately 3.727 km/h.