Two cyclists, alan and brian, are racing around oval track of length 450m on the same direction simultaneously from the same point. alan races around the track in 45 seconds before brian does and overtakes him every 9 minutes. what are their rates, in meters per minute?

Alan: 200 m/min

Brian: 150 m/min

Step-by-step 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.

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.

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