Chapter 0.1, Problem 17E

To determine

To find a linear equation that describes the position $d$ of the bicyclist as a function of $t$ .

Answer to Problem 17E

The equation of the line representing the relationship existing between the position $d$ of the bicyclist and time $t$ :

  $d=0.4t+2.6$ .

Explanation of Solution

Given:

Given that the position $d$ (in kilometers) of the bicyclist is a linear function of time $t$ (in minutes) and that $d=5$ when $t=6$ . Also, it is given that the bicyclist travels $2$ kilometers for every $5$ minutes.

Concept Used:

Slope of a line:

Suppose that the variables $d$ and $t$ keep a linear relationship for which $d$ is the dependent variable and $t$ is the independent variable.

Let $\Delta d$ represents the change in $d$ for some change $\Delta t$ in $t$ .

Then, the slope of the line:

  $m=\frac{\Delta d}{\Delta t}$ .

Point-Slope form of the equation of a line:

Given the slope $m$ of a line and a point of its passage $\left(d_1,t_1\right)$ . Then, the equation of the line is given to be:

  $d-d_1=m\left(t-t_1\right)$ .

Calculation:

It is given that the bicyclist travels $2$ kilometers for every $5$ minutes.

That is, the change in position $\Delta d$ of the bicyclist equals $2$ kilometers for a change in time: $\Delta t=5\text{minutes}$ .

Then, it follows that, the slope of the line representing the relationship between $d$ and $t$ :

  $\begin{array}{l}m=\frac{\Delta d}{\Delta t}\\ m=\frac{2}{5}\\ m=0.4\end{array}$ .

It is given that $d=5$ when $t=6$ .

That is, the line representing the relationship passes through the point $\left(d_1,t_1\right)=\left(5,6\right)$ .

Then, the equation of the line:

  $\begin{array}{c}d-d_1=m\left(t-t_1\right)\\ d-5=0.4\left(t-6\right)\\ d-5=0.4t-0.4\cdot 6\\ d-5=0.4t-2.4\\ d=0.4t-2.4+5\\ d=0.4t+2.6\end{array}$ .

Thus, it is found that:

  $d=0.4t+2.6$

That is, the equation of the line representing the relationship existing between the position $d$ of the bicyclist and time $t$ :

  $d=0.4t+2.6$ .

Conclusion:

The equation of the line representing the relationship existing between the position $d$ of the bicyclist and time $t$ :

  $d=0.4t+2.6$ .