Question

If they meet in 3 hours what is the rate of each cyclist

Answer 1

Speed (s) is defined as follows:

`s=(d)/(t)`

where d is distance and t is time.

Let's call x to the distance traveled by one of the cyclists. Combining the distance traveled by both cyclists, they rode 81 miles. Then, if one cyclist rode x miles, then the other one traveled (81 - x) miles, that is,

`\begin{gathered} d_1=81-x \\ d_2=x \\ d_1+d_2=81-x+x=81 \end{gathered}`

Where d1 and d2 are the distances traveled.

Substituting into the speed formula:

`\begin{gathered} s_1=(d_1)/(t)=(81-x)/(t) \\ s_2=(d_2)/(t)=(x)/(t) \end{gathered}`

They traveled 3 hours, then t = 3 in both equations.

We also know that one cyclist travels 5 miles/h slower than the other one. Therefore:

`\begin{gathered} s_1=s_2-5 \\ (81-x)/(t)=(x)/(t)-5 \\ (81-x)/(3)=(x)/(3)-5 \end{gathered}`

Solving for x:

`\begin{gathered} 3\cdot(81-x)/(3)=3\cdot((x)/(3)-5) \\ 81-x=x-15 \\ 81+15=x+x \\ 96=2x \\ (96)/(2)=x \\ 48=x \end{gathered}`

Substituting this result into the speed equations:

`\begin{gathered} s_1=(81-x)/(t) \\ s_1=(81-48)/(3) \\ s_1=(33)/(3) \\ s_1=11\frac{\text{ miles}}{hour} \end{gathered}`
`\begin{gathered} s_2=(x)/(t) \\ s_2=(48)/(3) \\ s_2=16(miles)/(hour) \end{gathered}`