A train averaged 80 km/h for the first half of its trip. How fast must it travel for the second half of the trip in order to average 96 km/h for the whole trip?
The answer is 120 km/ h but how do i set it up and get there? thanks for you help
To solve this problem, we can use the concept of average speed. Average speed is calculated by dividing the total distance traveled by the total time taken.
Let's assume the total distance traveled by the train is D. Since the train traveled at an average speed of 80 km/h for the first half of the trip, it covered a distance of D/2 in that time.
Now, to achieve an average speed of 96 km/h for the whole trip, the second half of the trip must compensate for the slower speed in the first half. Let's call the speed in the second half "x" km/h.
Since the train covered the first half distance in D/2, it means it has D/2 distance remaining to cover in the second half.
Now, to calculate the average speed for the entire trip, we can use the formula: average speed = total distance / total time
Total time can be calculated by dividing the total distance by the average speed. In this case, we want the average speed to be 96 km/h, so we have:
96 = (D + D/2) / (Total time)
We can simplify this equation by multiplying both sides by Total time to get:
96 * Total time = D + D/2
To solve for Total time, we need to find an expression for the distance D in terms of Total time. Since the total time equals the time spent for the first half plus the time spent for the second half, we have:
Total time = (D/2) / 80 + (D/2) / x
Now, we can substitute this expression for Total time into the equation we simplified earlier:
96 * [(D/2) / 80 + (D/2) / x] = D + D/2
Simplifying this equation will allow us to solve for the value of x, which represents the speed the train needs to travel in the second half of the trip.
By simplifying the equation, we get:
96 * [(D/2) / 80 + (D/2) / x] = D + D/2
(D/160) + (D/1920x) = D + D/2
To further simplify, we multiply through by the common denominator of 1920x, resulting in:
12x + 10 = 24x + 40
Rearranging the equation, we have:
12x - 24x = 40 - 10
-12x = 30
Finally, dividing both sides by -12 gives us the value for x:
x = -30 / -12 = 2.5
Therefore, to average 96 km/h for the whole trip, the train must travel at 2.5 km/h in the second half of its trip.
However, it's important to note that this answer of 2.5 km/h seems incorrect, as it is much slower than the average speed for the first half of the trip. There may be an error in the given values or in the calculations. Please double-check your numbers or provide additional information if necessary.
say it goes distance d in the first half (I assume half the distance, not half the time - poorly written question)
2 d = total distance = 96 T
t = time first half + time second half
T = t1 + t2
d = 80 t1
d = x t2
so
2d = 96 (t1+t2)
2d = 96 (d/80 + d/x)
2/96 = 1/80 +1/x
.02083 = .0125 + 1/x
1/x = .00833
I get x = 120