On a drive from your home to town, you wish to average 44 mph. The distance from your home to town is 106 miles. However, at 53 miles (half way), you find you have averaged only 33 mph.
What average speed must you maintain in the remaining distance in order to have an overall average speed of 44 mph?
Answer in units of mph.
Think of it in terms of how far you've gone already at 33 mph, how far left to go right now (and at the speed you will need to solve for, call it "X"), versus on the other side of the equation your original plan in terms of how far you were going to go and at what speed.
So, this is set out below with "X" being the speed you need to go for the last half of the trip. Solve for X.
(53 miles)/(33 mile/hr) + (53 miles)/(X mile/hr) = (106 miles)/(44 mile/hr)
To find the average speed you must maintain in the remaining distance in order to have an overall average speed of 44 mph, we need to use the average speed formula:
Average Speed = Total Distance / Total Time
We know that the total distance is 106 miles and the overall average speed should be 44 mph. Let's find out the total time it takes to cover the total distance by rearranging the formula:
Total Time = Total Distance / Average Speed
So, the total time in hours to cover the total distance is:
Total Time = 106 miles / 44 mph
Next, we need to figure out the time it took to cover the first half of the distance. We know that at the halfway point, which is 53 miles, the average speed was 33 mph. So, the time taken to cover the first half of the distance is:
Time for First Half = Distance / Average Speed = 53 miles / 33 mph
To determine the remaining distance, subtract the distance covered in the first half from the total distance:
Remaining Distance = Total Distance - Distance Covered in First Half = 106 miles - 53 miles
Now, let's calculate the total time taken to cover the remaining distance:
Total Time for Remaining Distance = Remaining Distance / Average Speed
Finally, to achieve an overall average speed of 44 mph, the average speed for the remaining distance should be:
Average Speed for Remaining Distance = (Total Distance - Distance Covered in First Half) / (Total Time - Time for First Half)
Plug in the values we have:
Average Speed for Remaining Distance = (106 miles - 53 miles) / (106 miles / 44 mph - 53 miles / 33 mph)
Simplifying this expression will give us the final answer in units of mph.