a heavy freight train made a trip of 360 miles, running part of the distance at 54 miles an hour and the rest at 48 miles an hour.if the entire trip took 7 hours,find the distance traveled at each rate.
To find the distance traveled at each rate, we can set up two equations based on the given information.
Let's assume that the distance traveled at 54 miles per hour is represented by variable 'x'.
The distance traveled at 48 miles per hour would then be represented by the expression '360 - x' since the total distance traveled is 360 miles.
Now, let's work on setting up the equations:
1. The time taken to travel 'x' miles at 54 miles per hour can be calculated using the formula: Time = Distance/Speed.
So, the time taken to travel 'x' miles at 54 miles per hour is: x/54.
2. The time taken to travel '360 - x' miles at 48 miles per hour is: (360 - x)/48.
The total time taken for the entire trip is given as 7 hours. So, we can write the equation:
x/54 + (360 - x)/48 = 7.
Now, let's solve for 'x' to find the distance traveled at 54 miles per hour:
Multiply through by the least common multiple of 54 (the coefficient of the first term) and 48 (the coefficient of the second term), which is 2592:
2592 * (x/54) + 2592 * ((360 - x)/48) = 2592 * 7.
After simplifying, we get:
48x + 54 * (360 - x) = 18144.
Now, expand and combine like terms:
48x + 19440 - 54x = 18144.
Subtract 19440 from both sides:
-6x = -1296.
Divide both sides by -6:
x = 216.
So, the distance traveled at 54 miles per hour is 216 miles.
To find the distance traveled at 48 miles per hour, substitute the value of 'x' into the expression '360 - x':
360 - 216 = 144.
Therefore, the distance traveled at 48 miles per hour is 144 miles.