1. A bicycle racer covers a 75 mile training route to prepare for an upcoming race. If the racer could increase his speed by 5 mph, he would complete the same course in 3/4 of the time. Find his average rate of speed.
2. Nina drove for 6 hours at an average rate of 50 mph. During the first 5 hours he averaged 52 mph. What was her average speed during the last hour of driving?
To solve both of these problems, we can use the formula:
Average Speed = Total Distance / Total Time
For problem 1:
Let's assume the racer's original speed is x mph. According to the problem, if he could increase his speed by 5 mph, he would complete the same course in 3/4 of the time.
This means that his original time would be 4/3 times the new time, or (4/3) * (75 / (x+5)).
We can set up the equation:
75 / x = (4/3) * (75 / (x+5))
Simplifying, we get:
3(x+5) = 4x
Solving for x, we find:
3x + 15 = 4x
15 = x
Therefore, his original speed was 15 mph.
For problem 2:
We know that Nina drove for 6 hours at an average rate of 50 mph. This means her total distance traveled is 6 * 50 = 300 miles.
During the first 5 hours, she averaged 52 mph. This means she traveled 52 * 5 = 260 miles during that time.
To find the distance she traveled during the last hour, we subtract the distance she traveled during the first 5 hours from the total distance:
Distance during the last hour = Total distance - Distance during first 5 hours
Distance during the last hour = 300 - 260
Distance during the last hour = 40 miles
Her average speed during the last hour is simply the distance traveled during the last hour divided by the time taken:
Average Speed = Distance during last hour / Time taken
Average Speed = 40 miles / 1 hour
Average Speed = 40 mph
Therefore, her average speed during the last hour of driving was 40 mph.