We have two variables: time, \(t\text{,}\) and distance, \(d\text{,}\) and the following data points:
\(~ t ~ \) |
\(d\) |
\(1\) |
\(189\) |
\(4\) |
\(360\) |
Gregor’s speed is the ratio of the distance he traveled to the time it took. The distance he traveled is the change in his odometer reading (from 189 miles to 360 miles), and the time it took is the change in the clock reading (from 1 pm to 4 pm).The units of this ratio are miles per hour.
In mathematics, we use the symbol \(\blert{\Delta}\) (delta) for change in. Thus
\begin{align*}
\text{distance traveled}\amp= \Delta d = 360-189 = 171 \text{ miles} \\
\text{time elapsed}\amp= \Delta t=4-1=3 \text{ hours}
\end{align*}
Gregor’s average speed is the ratio \(\dfrac{\text{distance traveled}}{\text{time elapsed}} =\dfrac{\Delta d}{\Delta t}\text{,}\) so
\begin{gather*}
\text{Speed}=\frac{\Delta d}{\Delta t} = \frac{171 \text{ miles}}{3 \text{ hours}}= 57 \text{ miles/hour}
\end{gather*}