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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 [tex]\( t \)[/tex], the amount of time Amir spent hiking during the second, more difficult part of the hike?

A. [tex]\( 3(2 - t) = 1.5t \)[/tex]
B. [tex]\( 3t = 1.5(2 - t) \)[/tex]
C. [tex]\( 3t + 1.5(2 - t) = 5 \)[/tex]
D. [tex]\( 3(2 - t) + 1.5t = 5 \)[/tex]



Answer :

GinnyAnswer
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.

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