Question

Suppose you have entered a 48-mile biathlon that consists of a run and a bicycle race. During your run, your average

velocity is 5 miles per hour, and during your bicycle race, your average velocity is 23 miles per hour. You finish the race
in 6 hours. What is the distance of the run? What is the distance of the bicycle race?

Answer 1

`\displaystyle \text{For the run:}`

`\displaystyle \text{Time taken} = \text{Distance/Velocity}`

`\displaystyle t_1 = (x)/(5)`

`\displaystyle \text{For the bicycle race:}`

`\displaystyle \text{Time taken} = \text{Distance/Velocity}`

`\displaystyle t_2 = (48 - x)/(23)`

`\displaystyle \text{Given that the total time for the race is 6 hours, we can write the equation:}`

`\displaystyle t_1 + t_2 = 6`

`\displaystyle \text{Substituting the expressions for } t_1 \text{ and } t_2, \text{ we get:}`

`\displaystyle (x)/(5) + (48 - x)/(23) = 6`

`\displaystyle 23x + 5(48 - x) = 6 * 115`

`\displaystyle \text{Simplifying further:}`

`\displaystyle 23x + 240 - 5x = 690`

`\displaystyle 18x = 450`

`\displaystyle x = (450)/(18)`

`\displaystyle x = 25`

`\displaystyle \text{Therefore, the distance of the run is 25 miles, and the distance of the bicycle race is } 48 - 25 = 23 \text{ miles.}`