a 10k race was held the first place finisher completed the 10km run in 29.5 minutes what was the average speed in miles per hour. In another race the person is given a head start and if they beat the first place finisher at the end the win $2,500 the chosen winner was given a 2.1 head start at what minimum average speed in miles per hour did he have run in order to win?
Speed = 10km/29.5min * 1mi/1.6km * 60min/h = 12.71 mi/h.
d1 = 10km/1.6 = 6.25 mi.
d1 = r1*t1 = 12.71*t1 = 6.25, t1 = 0.492 h.
d2 = (10-2.1)km/1.6 = 4.94 mi.
d2 = r2*t1 = r2*0.492 = 4.94, r2 = 10 mi/h. = min. speed to win.
To find the average speed in miles per hour (mph) for the first place finisher in the 10k race, we need to convert the distance from kilometers to miles and the time from minutes to hours.
1. Convert 10 kilometers to miles:
Since 1 kilometer is approximately 0.621371 miles, we can multiply 10 kilometers by 0.621371 to get the equivalent distance in miles:
10 km * 0.621371 = 6.21371 miles
2. Convert 29.5 minutes to hours:
Since 1 hour is equal to 60 minutes, we can divide 29.5 minutes by 60 to get the equivalent time in hours:
29.5 minutes / 60 = 0.49167 hours
3. Calculate the average speed:
To find the average speed, divide the distance (in miles) by the time (in hours):
Average Speed = Distance / Time = 6.21371 miles / 0.49167 hours
Average Speed ≈ 12.65 mph
Therefore, the average speed of the first place finisher in miles per hour is approximately 12.65 mph.
For the second race, let's calculate the minimum average speed in mph that the chosen winner needs to run in order to win.
1. Convert the head start distance of 2.1 kilometers to miles:
Using the conversion factor of approximately 0.621371, we can multiply 2.1 kilometers by 0.621371 to get the equivalent distance in miles:
2.1 km * 0.621371 = 1.3049391 miles (rounded to 7 decimal places)
2. Calculate the distance the chosen winner needs to run to win:
Since the chosen winner is given a 2.1 km head start and needs to beat the first-place finisher, their total distance would be the race distance (10 km) plus the head start distance (2.1 km):
Total distance = 10 km + 2.1 km = 12.1 km
3. Convert the total distance to miles:
Using the conversion factor of approximately 0.621371, we can multiply 12.1 kilometers by 0.621371 to get the equivalent distance in miles:
12.1 km * 0.621371 = 7.5201991 miles (rounded to 7 decimal places)
4. Calculate the minimum average speed needed to win:
To find the minimum average speed needed to win, divide the total distance (in miles) by the time (in hours), which is the same as the first place finisher's time (29.5 minutes) converted to hours:
Minimum Average Speed = Total Distance / Time = 7.5201991 miles / 0.49167 hours
Minimum Average Speed ≈ 15.29 mph
Therefore, the chosen winner needs to run at a minimum average speed of approximately 15.29 mph to win the race and receive the $2,500 prize.