Answer:
Alan: 200 m/min
Brian: 150 m/min
Explanation:
Let "a" and "b" represent Alan and Brian's rates in meters per minute, respectively. The rate at which Alan is lapping Brian is the difference in their rates:
a - b = (450 m)/(9 min) = 50 m/min
This lets us write Brian's rate in terms of Alan's as ...
a -50 = b
The time to complete one oval differs by 3/4 minute (45 seconds), so we have ...
time = distance/speed
450/b - 450/a = 3/4
Multiplying by 4ab/3 gives ...
600(a -b) = ab
Substituting from above, we can rewrite this as ...
600·50 = a(a -50)
a^2 -50a -30000 = 0 . . . . . quadratic rearranged to standard form
(a -200)(a +150) = 0 . . . . . . .factored
a = 200 or -150 . . . . only the positive solution is useful here
Alan's rate is 200 m/min; Brian's rate is 150 m/min.
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Check
It takes Alan 450/200 min = 2.25 min to complete one oval. It takes Brian 450/150 = 3 min to complete one oval. That is 3-2.25 = 0.75 min = 45 seconds longer than Alan.
After 9 minutes, Alan will have gone (200 m/min)·(9 min) = 1800 m = 4 laps, while Brian will have gone (150 m/min)·(9 min) = 1350 m = 3 laps. Hence Alan will overtake Brian at the 9-minute mark.