To determine the equation that shows the relationship between the time Jeff hikes, [tex]$t$[/tex] (in hours), and the distance he travels, [tex]$d$[/tex] (in miles), we first need to establish Jeff's hiking speed.
Jeff hiked for 2 hours and covered a distance of 5 miles. To find his speed, we divide the distance by the time:
[tex]\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{5 \text{ miles}}{2 \text{ hours}} = 2.5 \text{ miles per hour} \][/tex]
This means that for every hour Jeff hikes, he travels 2.5 miles.
Next, we set up the equation to represent the distance Jeff travels, [tex]$d$[/tex], based on the time he hikes, [tex]$t$[/tex]:
[tex]\[ d = 2.5t \][/tex]
This equation shows that the distance, [tex]$d$[/tex], is equal to 2.5 times the time, [tex]$t$[/tex].
Now, we need to determine whether the graph of this relationship will be continuous or discrete.
- A discrete graph consists of distinct, unconnected points. It applies to scenarios where the variables can only take certain values (e.g., whole numbers or counts of items).
- A continuous graph, on the other hand, consists of connected points extending in a smooth curve or a straight line. It represents situations where the variables can take any value within a certain range (e.g., measurements like time and distance).
Since Jeff can hike for any fraction of an hour and travel any corresponding distance (not limited to whole numbers or specific intervals), the graph of this relationship will be continuous.
Thus, the equation and the nature of the graph are:
[tex]\[ d = 2.5t, \quad \text{continuous} \][/tex]
