(a) The function describing Alice's distance from Bob's starting position in terms of t is:
Alice's distance = 3t
The function describing Bob's distance from Bob's starting position in terms of t is:
Bob's distance = 5(t-20)
Note that Bob starts running 20 seconds after Alice, so we subtract 20 from t to account for the time difference.
(b) To find the time at which Bob has run 50% farther than Alice, we need to find the value of t for which Bob's distance is 50% greater than Alice's distance.
Bob's distance = 5(t-20)
Alice's distance = 3t
Bob's distance is 50% greater than Alice's distance when:
5(t-20) = 1.5(3t)
Simplifying the equation:
5t - 100 = 4.5t
0.5t = 100
t = 200
Thus, Bob has run 50% farther than Alice after 200 seconds.