Find the speed of a train if an increase of 6kph lowers the time for a journey of 72 km by 10 minutes.
speed of train --- x km/h
time for 1st trip = 72/x
new speed = x+6
new time = 72/(x+6)
72/x - 72/(x+6) = 12/60 = 1/5
multiply by 5x(x+6)
360(x+6) - 360x = x(x+6)
360x + 2160 - 360x = x^2 + 6x
x^2 + 6x =2160
x^2 + 6x + 9 = 2160+9
(x+3)^2 = 2169
x+3 = ±√2169 = 46.57 , rejecting the negative answer
x = 43.57 km/h
just one observation: 10 minutes is not 1/5 hour
What is the time in the distance is 72 km in the speed is 6
kph
To find the speed of the train, we can use the formula:
Speed = Distance / Time
Let's consider the initial speed of the train as x kmph.
According to the given problem, an increase of 6 kmph lowers the time for a journey of 72 km by 10 minutes.
Step 1: Convert 10 minutes to hours.
There are 60 minutes in an hour, so 10 minutes is equal to 10/60 = 1/6 hours.
Step 2: Calculate the initial time taken for the journey.
The initial distance is 72 km, and the initial speed is x kmph.
Therefore, the initial time taken for the journey is 72 / x hours.
Step 3: Calculate the new time taken for the journey after the speed increase.
The new speed is (x + 6) kmph, and the new distance is still 72 km.
Therefore, the new time taken for the journey is 72 / (x + 6) hours.
According to the problem, the new time taken is 10 minutes (or 1/6 hours) less than the initial time.
Step 4: Set up the equation.
We can set up an equation based on the given information:
72 / x - 72 / (x + 6) = 1/6
Step 5: Solve the equation to find the value of x.
To solve this equation, we can multiply through by the common denominator, which is 6(x)(x + 6):
72(x + 6) - 72x = x(x + 6)
Distribute and simplify:
72x + 432 - 72x = x^2 + 6x
Bringing all the terms to one side, we get:
x^2 - 6x - 432 = 0
Now, we can factor or use the quadratic formula to solve for x. In this case, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values of a, b, and c into the formula, we get:
x = (6 ± √((-6)^2 - 4(1)(-432))) / (2(1))
Simplifying further:
x = (6 ± √(36 + 1728)) / 2
x = (6 ± √(1764)) / 2
x = (6 ± 42) / 2
This gives us two possible solutions for x:
1. x = (6 + 42) / 2 = 48 kmph
2. x = (6 - 42) / 2 = -18 kmph
However, since the speed of the train cannot be negative, we discard the second solution.
Therefore, the speed of the train is 48 kmph.