Exercise 58 Page 81

Barbara

Barbara began the race with a $3$ meter head start. Previously, we learned that the headstart of different participants correspond to the $y\text{-}$intercept of their line. We also know that she went $3$ meters every $2$ seconds. We can use this to create a second point and draw our line.

Mark

Mark began at the starting line, so we know that his $y\text{-}$intercept must correspond to the origin. We also know that he finished the entire race, $20$ meters, in $5$ seconds. This means that a second point on the line is going to be $(5,20).$

Carlos

For Carlos, we were already given the equation of his line, $y=\frac{5}{2}x+1.$ This tell us that his $y\text{-}$intercept is $1$ and he received a $1$ meter head start. We also know that he rode $5$ meters every $2$ seconds.

Combined Graphs

We can combine all three of these lines on the same coordinate plane to compare the three participants.

Barbara

A line in slope-intercept form is written in the following format. In the equation, $m$ is the slope and $b$ is the $y\text{-}$intercept. We already know that Barbara's $y\text{-}$intercept is $3.$ We can find the slope of the line by thinking about what the changes in vertical and horizontal distance are for this situation. The final equation for Barbara's line is as follows.

Mark

Using the same process, we can find the equation for Mark's line. We know that his $y\text{-}$intercept is $0.$ Let's also find the slope. Therefore, the final equation for Mark is the following.

In this formula $d$ is distance, $r$ is rate, and $t$ is the time spent traveling. Based on the slope of his equation, we know that he pedaled $5$ meters every $2$ seconds. Let's use this information to solve for his speed. Carlos's speed was $2.5$ m/s.

Carlos passed Barbara after $2$ seconds, at the $6$ meter mark. We can confirm this algebraically by setting their equations equal to one another. Now that we have found the time at which they were equal distances from the starting line, we can confirm the distance when that occurred. They were both at the $6$ meter mark after $2$ seconds.