Let's solve the problem step-by-step.
1. Define the given information:
- The total distance Amir hiked is 5 miles.
- The total time he took to hike is 2 hours.
- On the first part of the hike, Amir's speed was 3 miles per hour.
- On the second part of the hike, his speed was 1.5 miles per hour.
2. Introduce the variable:
- Let [tex]\( t \)[/tex] be the time (in hours) Amir spent hiking during the second part of the hike.
3. Determine the time spent on the first part of the hike:
- Since the total hiking time is 2 hours, the time spent on the first part is [tex]\( 2 - t \)[/tex] hours.
4. Write the equation for the distance covered in each part:
- Distance covered on the first part of the hike = (speed on the first part) [tex]\( \times \)[/tex] (time spent on the first part) = [tex]\( 3 \times (2 - t) \)[/tex] miles.
- Distance covered on the second part of the hike = (speed on the second part) [tex]\( \times \)[/tex] (time spent on the second part) = [tex]\( 1.5 \times t \)[/tex] miles.
5. Combine the distances to match the total distance:
- The distance covered on the first part plus the distance covered on the second part should equal the total distance:
[tex]\[
3(2 - t) + 1.5t = 5
\][/tex]
Therefore, the equation that can be used to find [tex]\( t \)[/tex], the time Amir spent hiking during the second, more difficult part of the hike, is:
[tex]\[
3(2 - t) + 1.5t = 5
\][/tex]
This matches the fourth option.
