Question

Suppose you have entered a 178-mile biathlon that consists of a run and a bicycle race. During your run, your average velocity is 7 miles per hour, and during your bicycle race, your average velocity is 30 miles per hour. You finishthe race in 9 hours. What is the distance of the run? What is the distance of the bicycle race?The distance of the run ismiles

Answer 1

The following information is provided in the question:

$\begin{gathered} Total\text{ }Distance=178\text{ }miles \\ Velocity\text{ }of\text{ }Run=7mph \\ Velocity\text{ }of\text{ }Bicycling=30mph \\ Time\text{ }Taken=9\text{ }hours \end{gathered}$

Recall the formula relating distance, speed, and time:

$s=\frac{d}{t}$

Let the distance for the run be x. This means that the distance for the bicycle race will be 178 - x.

Were we to use the provided information to calculate the time for each part of the race, we would have:

$\begin{gathered} time\text{ }of\text{ }run=\frac{x}{7} \\ time\text{ }of\text{ }bicycle\text{ }race=\frac{178-x}{30} \end{gathered}$

Since the total time is 9, we have:

$\frac{x}{7}+\frac{178-x}{30}=9$

Solving, we have:

$\begin{gathered} \mathrm{Multiply\:by\:LCM=}210 \\ \frac{x}{7}\cdot \:210+\frac{178-x}{30}\cdot \:210=9\cdot \:210 \\ 30x+7\left(-x+178\right)=1890 \\ Simplify \\ 30x-7x+1246=1890 \\ 30x-7x+1246=1890 \\ 23x=644 \\ \mathrm{Divide\:both\:sides\:by\:}23 \\ x=28 \end{gathered}$

Therefore, the distance for the bicycle race will be:

$\Rightarrow178-28=150$

The distance of the run is 28 miles.

The distance of the bicycle race is 150 miles.