Amy wants to travel from elm town to lakeside at an average speed of 50 miles per hour. She travels half the distance and finds out her average speed has been on 25 miles per hour. How fast must she drive for the rest of the trip to average 50 miles per H our for the whole trip.?
I say 75 miles per hour
let's call the distance d
so the desired time for Amy's trip is ... d / 50
her time at the halfway point is ... (d / 2) / 25 , which equals ... d / 50
she has run out of time to achieve her desired average speed
To calculate the speed that Amy must drive for the rest of the trip in order to average 50 miles per hour for the entire trip, we can use the formula:
Average Speed = Total Distance / Total Time
Let's assume the total distance between Elm Town and Lakeside is "D" miles. Since Amy travels half the distance at an average speed of 25 miles per hour, she covers (D/2) miles at this speed.
Let's denote the remaining distance as (D/2) miles. We need to find the speed Amy must drive for this remaining distance to have an average speed of 50 miles per hour for the whole trip.
Let's calculate the total time it takes for Amy to travel the first half of the distance:
Time taken for first half = Distance / Speed = (D/2) / 25 = D/50 hours
Now, let's calculate the time it takes to travel the remaining distance at the desired average speed of 50 miles per hour:
Time taken for the second half = Remaining Distance / Speed(x) = (D/2) / x
To have an average speed of 50 miles per hour, the total time for the entire trip remains the same. So we can equate the two time expressions:
D/50 = (D/2) / x
Simplifying the equation:
D * x = 50 * (D/2)
x = 50 * (D/2) / D
x = 25
Hence, Amy must drive at a speed of 25 miles per hour for the remaining distance to average 50 miles per hour for the whole trip.
To solve this problem, we need to 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 between Elm Town and Lakeside is D miles. Amy travels half the distance, which means she travels D/2 miles.
Given that Amy's average speed for the first half of the trip is 25 miles per hour, we can calculate the time taken for the first half of the trip by dividing the distance traveled by the average speed:
Time taken for the first half of the trip = (D/2) / 25 = D/50 hours.
Now Amy wants to have an average speed of 50 miles per hour for the whole trip. Let's assume she needs to drive at a speed of V miles per hour for the rest of the trip.
The distance for the second half of the trip is also D/2 miles. To maintain an average speed of 50 miles per hour for the whole trip, the total time taken (including the first half) should be:
Total time taken = Time taken for the first half + Time taken for the second half
Total time taken = D/50 + (D/2) / V
The total time taken is calculated by dividing the total distance by the average speed:
Total time taken = D / 50 mph + D / V mph
To find the average speed for the whole trip, we divide the total distance by the total time taken:
Average speed = Total distance / Total time taken
50 mph = D / (D/50 + D/V)
Now, we can solve for V by multiplying both sides of the equation by (D/50 + D/V):
50 mph * (D/50 + D/V) = D
D + DV/50 = D
DV/50 = 0
DV = 0
The equation DV = 0 indicates that there is no solution for V, as it would result in an average speed of 0 mph. Therefore, it is not possible for Amy to average a speed of 50 mph for the whole trip if her average speed for the first half is 25 mph.