### Home > AC > Chapter 9 > Lesson 9.2.2 > Problem9-43

9-43.

During a race, Bernie ran $9$ meters every $4$ seconds, while Barnaby ran $2$ meters every second and got a $10$-meter head start. If the race was $70$ meters long, did Bernie ever catch up with Barnaby? If so, when? Justify your answer.

Define variables and write an equation for Bernie and an equation for Barnaby.
$x =$ time in seconds
$y =$ distance in meters

$\text{Bernies's rate} = \frac{9\text{ meters}}{4\text{ seconds}}$

$\text{Wendel's rate} = \frac{2\text{ meters}}{1\text{ second}}$ $10 \; \text{meter head start} = \text{starting point}$

$y=\frac{9}{4}x$

$y = 2x + 10$

To catch up, their equations would need to be equal.

$\frac{9}{4}x=2x+10$

Solve for $x$.

$x=40 \; \text{seconds}$

Substitute $x$ into one of the original equations and find $y$.
If the distance $\left(y\right)$ is less than $70$, Bernie caught up with Barnaby.

$y = 90 \; \text{meters}$
Bernie can't catch up with Barnaby in $70$ meters.

Use the eTool below to graph the equations for Bernie and Barnaby.
Look at your graph to determine if Bernie ever caught up with Barnaby and if so, when.
Click on the link at right for the full eTool version: 9-43 HW eTool (Desmos)