Imagine that you are driving 150 miles and you want your average speed for the trip to be 60 miles per hour.Imagine that you drive the first half (by distance) of the trip at 50 mph. How fast must you drive on the second half to make your average speed for the whole trip 60 mph? (Hint: The answer is NOT 70 mph.)
If it's not 70mph, what else could it possibly be?
You want your total time to be
150/60 = 5/2 hours
Since time = distance/speed, you need
75/50 + 75/x = 5/2
x = 75
To determine the speed you need to drive on the second half of your trip to achieve an average speed of 60 mph, we need to use a weighted average formula.
First, we can calculate the time it takes to drive the first half of the trip at a speed of 50 mph. We divide the distance (75 miles, as it's half of 150 miles) by the speed:
Time = Distance / Speed = 75 miles / 50 mph = 1.5 hours
Now, to find the speed you need to drive on the second half to maintain an overall average speed of 60 mph, we can use the formula:
Average Speed = Total Distance / Total Time
Let's denote the time it takes to drive the second half as T2 and the speed on the second half as S2.
The remaining distance for the second half of the trip is also 75 miles. So the total distance is 150 miles.
Average Speed = 60 mph
Total Distance = 150 miles
Total Time = 1.5 hours (from the first half) + T2 (unknown)
Now we can rearrange the formula to solve for T2:
Average Speed = Total Distance / Total Time
60 mph = 150 miles / (1.5 hours + T2)
To solve for T2, we multiply both sides of the equation by (1.5 hours + T2):
60 mph * (1.5 hours + T2) = 150 miles
Expanding this equation:
90 miles + 60 mph * T2 = 150 miles
Subtracting 90 miles from both sides:
60 mph * T2 = 60 miles
Finally, divide both sides by 60 mph:
T2 = 1 hour
So, to achieve an average speed of 60 mph for the entire trip, you must drive at a speed of 60 mph on the second half of the trip.