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 □ miles.

Explanation:

Step1: Let the time for running be $t_1$ hours and the time for cycling be $t_2$ hours. Let the distance of the run be $d_1$ and the distance of the bicycle - race be $d_2$. We know that $d_1 + d_2=146$ miles and $t_1 + t_2 = 9$ hours. Also, the speed of running $v_1 = 9$ miles per hour and the speed of cycling $v_2=22$ miles per hour. Using the formula $d = vt$, we have $d_1=v_1t_1 = 9t_1$ and $d_2=v_2t_2=22t_2$.

Step2: Since $t_2=9 - t_1$, we substitute $d_1 = 9t_1$, $d_2 = 22(9 - t_1)$ into $d_1 + d_2=146$. So, $9t_1+22(9 - t_1)=146$.

Expand the equation: $9t_1+198 - 22t_1=146$.
Combine like - terms: $9t_1-22t_1=146 - 198$.
$- 13t_1=-52$.

Step3: Solve for $t_1$:

Divide both sides of the equation $-13t_1=-52$ by $-13$. We get $t_1 = 4$ hours.

Step4: Calculate the distance of the run $d_1$:

Since $d_1 = v_1t_1$ and $v_1 = 9$ miles per hour, $t_1 = 4$ hours, then $d_1=9\times4 = 36$ miles.

Answer:

36