Question

suppose you have entered a 146 - mile biathlon that consists of a run and a bicycle race. during your run, your average velocity is 9 miles per hour, and during your bicycle race, your average velocity is 22 miles per hour. you finish the race in 9 hours. what is the distance of the run? what is the distance of the bicycle race? the distance of the run is 36 miles. the distance of the bicycle race is □ miles.

Explanation:

Step1: Calculate the distance of the run

We know that speed = distance/time, so distance = speed×time. The speed during the run is 9 miles per hour and the total time of the race is 9 hours. Let the time of the run be $t_{run}$ and the time of the bicycle - race be $t_{bike}$, and $t_{run}+t_{bike}=9$ hours. The speed during the run $v_{run}=9$ miles per hour. The distance of the run $d_{run}=v_{run}\times t_{run}$. We also know that the total distance of the bi - athlon $D = 146$ miles, and the speed during the bicycle race $v_{bike}=22$ miles per hour. Let the distance of the run be $d_{run}$ and the distance of the bicycle race be $d_{bike}$. So $D=d_{run}+d_{bike}$, and $d_{run}=v_{run}\times t_{run}$, $d_{bike}=v_{bike}\times t_{bike}$. Since $t_{run}+t_{bike}=9$, we can express $t_{bike}=9 - t_{run}$. Then $146=9t_{run}+22(9 - t_{run})$.
Expand the right - hand side: $146=9t_{run}+198-22t_{run}$.
Combine like terms: $146=198 - 13t_{run}$.
Rearrange to solve for $t_{run}$: $13t_{run}=198 - 146$, $13t_{run}=52$, $t_{run}=4$ hours.
So the distance of the run $d_{run}=v_{run}\times t_{run}=9\times4 = 36$ miles.

Step2: Calculate the distance of the bicycle race

The total distance of the bi - athlon is 146 miles and the distance of the run is 36 miles.
Using the formula $d_{bike}=D - d_{run}$, we substitute $D = 146$ miles and $d_{run}=36$ miles.
$d_{bike}=146 - 36=110$ miles.

Answer:

The distance of the run is 36 miles. The distance of the bicycle race is 110 miles.