To solve the problem of determining how long Xóchitl needs to pedal to catch up to Cowessess, let's break it down step-by-step.
1. Understanding the Initial Distance Covered by Cowessess:
- Cowessess starts pedaling 12 minutes before Xóchitl and maintains a constant rate of [tex]\(10 \frac{ km }{ h }\)[/tex].
- First, convert the 12 minutes into hours:
[tex]\[
\text{Time}_\text{Cowessess} = \frac{12 \text{ minutes}}{60 \text{ minutes/hour}} = 0.2 \text{ hours}
\][/tex]
- Determine how far Cowessess has pedaled during these 12 minutes using the formula for distance [tex]\((\text{Distance} = \text{Rate} \times \text{Time})\)[/tex]:
[tex]\[
\text{Distance}_\text{Cowessess} = 10 \frac{ km }{ h } \times 0.2 \text{ hours} = 2 \text{ km}
\][/tex]
2. Setting Up the Equation for Xóchitl to Catch Up:
- Xóchitl starts pedaling at a rate of [tex]\(18 \frac{ km }{ h }\)[/tex].
- Both will travel the same distance to meet, so let’s denote the additional time Xóchitl needs to pedal as [tex]\( \text{time}_x \)[/tex].
- During this time, Xóchitl covers:
[tex]\[
\text{Distance}_\text{Xóchitl} = 18 \frac{ km }{ h } \times \text{time}_x
\][/tex]
- During the same time, Cowessess continues to pedal at her constant rate, covering:
[tex]\[
\text{Distance}_\text{Cowessess}_\text{additional} = 10 \frac{ km }{ h } \times \text{time}_x
\][/tex]
- The total distance Cowessess would have traveled when Xóchitl catches up is the initial distance plus the additional distance:
[tex]\[
\text{Total Distance}_\text{Cowessess} = 2 \text{ km} + 10 \frac{ km }{ h } \times \text{time}_x
\][/tex]
3. Equating the Distances:
- To find when Xóchitl catches up, set the distance covered by Xóchitl equal to the total distance covered by Cowessess:
[tex]\[
18 \frac{ km }{ h } \times \text{time}_x = 2 \text{ km} + 10 \frac{ km }{ h } \times \text{time}_x
\][/tex]
4. Solving for Time:
- Isolate [tex]\(\text{time}_x\)[/tex] by bringing the terms involving [tex]\(\text{time}_x\)[/tex] to one side:
[tex]\[
18 \frac{ km }{ h } \times \text{time}_x - 10 \frac{ km }{ h } \times \text{time}_x = 2 \text{ km}
\][/tex]
- Simplify:
[tex]\[
(18 \frac{ km }{ h } - 10 \frac{ km }{ h }) \times \text{time}_x = 2 \text{ km}
\][/tex]
[tex]\[
8 \frac{ km }{ h } \times \text{time}_x = 2 \text{ km}
\][/tex]
- Divide both sides by [tex]\(8 \frac{ km }{ h }\)[/tex]:
[tex]\[
\text{time}_x = \frac{2 \text{ km}}{8 \frac{ km }{ h }} = 0.25 \text{ hours}
\][/tex]
Therefore, Xóchitl would have to pedal for 0.25 hours (or 15 minutes) to catch up to Cowessess's distance.
