A cyclist leaves his training base for a morning workout, riding at the rate of 18 mph. One hour later, his support staff leaves the base in a car going 45 mph in the same direction. How long will it take the support staff to catch up with the cyclist?

same distance, 3/4 hr difference.

d=18t
d=45(t-3/4)
set them equal, solve for t, the total time the cyclist biked. Then, solve for t-3/4

2/3

To find out how long it will take the support staff to catch up with the cyclist, we need to determine the distance the cyclist will have covered by that time.

Let's assume the time it takes for the support staff to catch up with the cyclist is represented by 't' (in hours). Since the cyclist has a one-hour head start, the cyclist will have traveled for t + 1 hours.

The formula to calculate distance is: Distance = Speed × Time.

For the cyclist, the distance traveled is: Distance_cyclist = Speed_cyclist × Time_cyclist.
Given that the speed of the cyclist is 18 mph, and he rides for t + 1 hours, we have: Distance_cyclist = 18(t + 1) miles.

For the support staff, the distance traveled is: Distance_staff = Speed_staff × Time_staff.
Given that the speed of the support staff is 45 mph, and they travel for t hours, we have: Distance_staff = 45t miles.

Since the distance traveled by both the cyclist and the support staff when the support staff catches up with the cyclist will be the same, we can set the two distances equal to each other:

18(t + 1) = 45t

Let's solve this equation for 't':

18t + 18 = 45t

Subtracting 18t from both sides:

18 = 45t - 18t

Simplifying:

18 = 27t

Dividing both sides by 27:

t = 18/27

Simplifying:

t = 2/3

Thus, it will take the support staff approximately 2/3 hours (or 40 minutes) to catch up with the cyclist.