Answered

It took Amir 2 hours to hike 5 miles. On the first part of the hike, Amir averaged 3 miles per hour. For the second part of the hike, the terrain was more difficult, so his average speed decreased to 1.5 miles per hour.



Which equation can be used to find $t$, the amount of time Amir spent hiking during the second, more difficult part of the hike?


A$3(2-t)=1.5 t$
B$3 t=1.5(2-t)$
C$3 t+1.5(2-t)=5$
D$3(2-t)+1.5 t=5$

Verified Answer :

  • Let \(t\) be the time in hours for the second, more difficult part of the hike, and \((2-t)\) be the time for the first part.
  • Express the distance of each part as the product of speed and time: \(3(2-t)\) and \(1.5t\).
  • Sum these distances to equal the total distance of \(5\) miles.
  • The correct equation is \(\boxed{3(2-t) + 1.5t = 5}\).

Explanation

  1. Analysis of the Problem

To find the time \(t\) spent on the second part of the hike, we begin by analyzing the data provided:

  • Total distance hiked: \(5\) miles.
  • Total time spent hiking: \(2\) hours.
  • Speed on the first part of the hike: \(3\) miles per hour.
  • Speed on the second, more difficult part: \(1.5\) miles per hour.
  • Time spent on the second part: \(t\) hours.

Consequently, the time spent on the first part of the hike is the remaining time, which is \((2 - t)\) hours.

  1. Formulating the Equation

We know that the relationship between distance, speed, and time is given by the formula:
\[\text{Distance} = \text{Speed} \times \text{Time}\]
Applying this to each part of the hike:

  • Distance of the first part: \(\text{Speed}_1 \times \text{Time}_1 = 3 \times (2 - t)\)
  • Distance of the second part: \(\text{Speed}_2 \times \text{Time}_2 = 1.5 \times t\)

The total distance is the sum of the distances of the two parts:
\[\text{Distance}_1 + \text{Distance}_2 = 5\]
Substituting the expressions derived above, we get:
\[3(2 - t) + 1.5t = 5\]

  1. Identifying the Correct Equation

Comparing the derived equation \(3(2 - t) + 1.5t = 5\) to the given options, we see that it matches the fourth option. This equation correctly represents the sum of the distances of the two hike segments equaling the total distance covered by Amir.

Examples

Understanding how to model distance and time problems with equations is essential for everyday planning and logistics. For instance, if you are planning a trip and know your average speeds on different terrains—such as walking on a flat path versus hiking up a steep hill—you can use similar equations to estimate how long each segment of your journey will take. This helps in setting realistic arrival times and managing your energy effectively during outdoor activities.