Alice and Bob go for a run in the local park. Alice runs at 3 m/s. Bob starts from the same point as Alice, but he starts 20 seconds after Alice. Bob runs at a rate of 5 m/s.
(a) Let t be the number of seconds that have elapsed since Bob started running. Find functions describing Alice's and Bob's distance in meters from Bob's starting position in terms of t.
(b) How many seconds after Bob starts running has he run 50% farther than Alice
(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.