Determining an Equation and Type of Graph
Jeff hiked for 2 hours and traveled 5 miles. If he continues at the same pace, which equation will show the relationship between the time, [tex]$t$[/tex], in hours he hikes to distance, [tex]$d$[/tex], in miles? Will the graph be continuous or discrete?
A. [tex]$d=0.4 t$[/tex], discrete
B. [tex]$d=0.4 t$[/tex], continuous
C. [tex]$d=2.5 t$[/tex], discrete
D. [tex]$d=2.5 t$[/tex], continuous
Answer (1)
Calculate Jeff's speed: s p ee d = 2 5 = 2.5 miles per hour .
Write the equation relating distance and time: d = 2.5 t .
Determine that the graph is continuous because time can take on any value.
The equation is d = 2.5 t and the graph is continuous, so the final answer is d = 2.5 t , continuous .
Explanation
Calculate Jeff's Speed First, we need to determine Jeff's speed. We know he hiked 5 miles in 2 hours. We can calculate his speed by dividing the distance he traveled by the time it took him to travel that distance.
Determine Jeff's Speed Jeff's speed is calculated as follows: s p ee d = t im e d i s t an ce = 2 hours 5 miles = 2.5 miles per hour So, Jeff's speed is 2.5 miles per hour.
Write the Equation Now that we know Jeff's speed, we can write an equation that relates the distance, d , he hikes to the time, t , he hikes. The equation is: d = s p ee d × t im e d = 2.5 t
Determine if the Graph is Continuous or Discrete Next, we need to determine if the graph of this equation is continuous or discrete. Since Jeff can hike for any amount of time (e.g., 2.1 hours, 2.25 hours, etc.), the time, t , can take on any value. This means the graph is continuous.
Final Answer Therefore, the equation that shows the relationship between the time, t , in hours Jeff hikes to the distance, d , in miles is d = 2.5 t , and the graph will be continuous.
Examples
Understanding the relationship between time and distance is crucial in many real-life scenarios. For instance, if you're planning a road trip, knowing your average speed helps you estimate how long it will take to reach your destination. Similarly, in sports, athletes use this relationship to track their performance and adjust their training accordingly. This concept also applies to fields like logistics and transportation, where efficient route planning relies on accurately predicting travel times based on speed and distance.
