On a drive from your home to town, you
wish to average 48 mph. The distance from
your home to town is 106 miles. However,
at 53 miles (half way), you find you have
averaged only 36 mph.
What average speed must you maintain in
the remaining distance in order to have an
overall average speed of 48 mph?
To average 48 mi/hr, you need to cover 106 miles in 2.208 hours.
The first half took 53/36 = 1.472 hours
So, how fast do you need to go to cover the other 53 miles in (2.208-1.472) hours?
To find the average speed you need to maintain in the remaining distance in order to have an overall average speed of 48 mph, we can use the formula:
Average Speed = Total Distance / Total Time
Let's break down the information given:
Distance from home to town = 106 miles
Distance traveled halfway = 53 miles
Average speed in the first half = 36 mph
To find the remaining distance, we subtract the distance traveled halfway from the total distance:
Remaining Distance = Total Distance - Distance Traveled Halfway
Remaining Distance = 106 miles - 53 miles
Remaining Distance = 53 miles
Now, let's calculate the total time taken in the first half of the trip:
Time taken in the first half = Distance Travelled / Average Speed
Time taken in the first half = 53 miles / 36 mph
To find the average speed you need to maintain in the remaining distance, we need to calculate the total time taken for the entire trip:
Total Time = Time taken in the first half + Time taken in the second half
To calculate the time taken in the second half, we can use the formula:
Time taken in the second half = Remaining Distance / Average Speed in the second half
Since we want the overall average speed of the entire trip to be 48 mph, we can set up the equation:
48 mph = Total Distance / Total Time
Now, let's solve for the average speed in the second half:
48 mph = (106 miles) / (Time taken in the first half + Remaining Distance / Average Speed in the second half)
By substituting the values we know:
48 mph = 106 miles / (53 miles / Average Speed in the second half + 36 mph)
Now, let's solve for Average Speed in the second half:
48 mph = 106 miles / (53 miles / Average Speed in the second half + 36 mph)
To simplify the equation, we can multiply both sides by (53 miles / Average Speed in the second half + 36 mph):
48 mph * (53 miles / Average Speed in the second half + 36 mph) = 106 miles
Simplifying further, we can distribute the 48 mph:
48 mph * 53 miles / Average Speed in the second half + 48 mph * 36 mph = 106 miles
Now, let's solve for Average Speed in the second half:
2514 / Average Speed in the second half + 1728 = 106
Subtracting 1728 from both sides:
2514 / Average Speed in the second half = 106 - 1728
Combining the numbers on the right side:
2514 / Average Speed in the second half = -1622
Taking the reciprocal of both sides of the equation:
Average Speed in the second half = 2514 / -1622
Finally, let's calculate the average speed in the second half:
Average Speed in the second half ≈ -1.551 mph
Therefore, in order to have an overall average speed of 48 mph, you would need to maintain an average speed of approximately -1.551 mph in the remaining distance. However, since negative speeds are not possible, it is not possible to achieve an overall average speed of 48 mph in this scenario.