Let's break down and analyze the problem step by step:
### Step 1: Convert Distance and Speed into Improper Fractions
1. Trail Distance:
- Santana wants to hike the South Boundary Trail which is [tex]\( 21 \frac{2}{5} \)[/tex] miles long.
- Convert the mixed number into an improper fraction:
[tex]\[
21 \frac{2}{5} = 21 + \frac{2}{5} = \frac{105}{5} + \frac{2}{5} = \frac{107}{5} \text{ miles}
\][/tex]
2. Hiking Speed:
- Santana can hike [tex]\( 2 \frac{1}{2} \)[/tex] miles in 60 minutes.
- Convert the mixed number into an improper fraction:
[tex]\[
2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \text{ miles in 60 minutes}
\][/tex]
### Step 2: Convert the Speed from per 60 Minutes to per Minute
- Since [tex]\( 2 \frac{1}{2} \)[/tex] miles is covered in 60 minutes, find how far she hikes in one minute:
[tex]\[
\text{Hiking speed per minute} = \frac{5}{2} \div 60 = \frac{5}{2} \times \frac{1}{60} = \frac{5}{120} = \frac{1}{24} \text{ miles per minute}
\][/tex]
### Step 3: Calculate the Time to Hike the Entire Trail
- To find out how long it will take her to hike [tex]\( \frac{107}{5} \)[/tex] miles at [tex]\( \frac{1}{24} \)[/tex] miles per minute:
[tex]\[
\text{Time in minutes} = \text{Distance} \div \text{Speed} = \frac{107}{5} \div \frac{1}{24} = \frac{107}{5} \times 24 = \frac{107 \times 24}{5} = \frac{2568}{5} = 513.6 \text{ minutes}
\][/tex]
### Step 4: Convert Time from Minutes to Hours and Minutes
- First, convert 513.6 minutes into hours and remaining minutes:
[tex]\[
\text{Hiking Time in Hours} = \left\lfloor \frac{513.6}{60} \right\rfloor = \left\lfloor 8.56 \right\rfloor = 8 \text{ hours}
\][/tex]
[tex]\[
\text{Remaining Minutes} = 513.6 - (8 \times 60) = 513.6 - 480 = 33.6 \text{ minutes}
\][/tex]
### Diagram
To support the understanding, consider drawing a simple diagram showing the start and end of the trail, breaking it down into multiple 2.5 mile segments and indicating how time is calculated progressively.
Time Calculation Diagram:
```
|------|------|------|------|------|------|------|------|------|------|------|------|
0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0)
0 60 120 180 240 300 360 420 480 540 600 660 720)
```
Each segment represents [tex]\( 2 \frac{1}{2} \)[/tex] miles requiring 60 minutes.
### Conclusion
The total time it will take Santana to hike the entire [tex]\( 21 \frac{2}{5} \)[/tex] mile South Boundary Trail is:
- Time in minutes: 513.6 minutes
- Time in hours and minutes: 8 hours and 33.6 minutes
