a cyclist completes a 150-mile race in a certain amount of time.she completes another 150-mile race a month later,but this time it takes an hour longer to cover the same distance and her average speed is 5 mph less than in the first race.find the average speed for the cyclist during her first race
30mph
To find the average speed of the cyclist during her first race, we can break down the problem into two steps:
Step 1: Determine the time taken for the second race.
Step 2: Calculate the average speed for the first race.
Step 1: Determine the time taken for the second race.
Let's call the time taken for the first race as t hours. Since the cyclist completed the 150-mile race in a certain amount of time, we know that the distance divided by time gives us the speed.
So, for the first race, the speed = 150 miles / t hours.
In the second race, the cyclist takes an hour longer to complete the same distance. Therefore, the time for the second race would be (t + 1) hours.
Step 2: Calculate the average speed for the first race.
Given that the average speed for the second race is 5 mph less than that of the first race, we can set up the following equation:
150 miles / (t + 1) hours = (150 miles / t hours) - 5 mph
Now, we can solve this equation to find the value of t.
First, let's cross multiply:
150 miles * t = (150 miles - 5 mph) * (t + 1) hours
Expanding the equation:
150t = (150t) + (150 - 5t) hours
150t = 150t + 150 - 5t
150t - 150t = 150 - 5t
0 = 150 - 5t
5t = 150
t = 150 / 5
t = 30
The time taken for the first race is 30 hours.
Now that we know the time taken, we can calculate the average speed for the first race:
Average speed = Distance / Time
Average speed = 150 miles / 30 hours
Average speed = 5 miles per hour (mph)
Therefore, the average speed for the cyclist during her first race is 5 mph.