A summer biathalon-style race requires running a certain distance, and then biking a distance 2 miles more than the length of the run. One athlete has a running speed of 6 miles per hour, and a biking speed of 10 miles per hour, and completes the event in 90 minutes. How long, in miles, is each segment of the race

The run is 4.875 miles long, while, the bike is 6.875 miles long.

Step-by-step explanation:

Let d = running distance and d' = biking distance.

Given that d' = d + 2 and v = running speed = 6 mph and v' = biking speed = 10 mph.

Since time, t = distance/speed, running time, t = running distance/running speed = d/6 and biking time, t' = biking distance/biking time = d'/10 = (d + 2)/10.

Since the total time, t" = 90 minutes = 1.5 hours,

t + t' = t"

So, d/6 + (d + 2)/10 = 1.5

multiplying through by the L.C.M, 30, we have

30 × d/6 + 30 ×(d + 2)/10 = 30 × 1.5

5d + 3(d + 2) = 45

expanding the brackets, we have

5d + 3d + 6 = 45

collecting like terms, we have

8d = 45 - 6

8d = 39

d = 39/8

d = 4.875 miles

d' = d + 2 = 4.875 + 2 = 6.875 miles

So, the run is 4.875 miles long, while, the bike is 6.875 miles long.

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