Wyatt and Kirk are both swimming across Crystal Lake. Kirk swims at a rate of 3 kilometers per hour, and Wyatt swims at a rate of 3.8 kilometers per hour. When Wyatt started, Kirk had already covered 0.2 kilometers.

How long will it take Wyatt to catch up to Kirk?

To determine how long it will take Wyatt to catch up to Kirk, we need to consider the relative speeds of both swimmers and the distance Kirk has ahead of Wyatt.

### Given:
- Kirk's speed = 3 km/h
- Wyatt's speed = 3.8 km/h
- Kirk's head start = 0.2 km

### Setting up the problem:
Let \( t \) be the time (in hours) it takes for Wyatt to catch up to Kirk.

During this time \( t \):
- The distance Kirk covers is \( 3t \) kilometers (since he swims at 3 km/h).
- The distance Wyatt covers is \( 3.8t \) kilometers (since he swims at 3.8 km/h).

For Wyatt to catch up to Kirk, the distance Wyatt swims must equal the distance Kirk swims plus the 0.2 km head start that Kirk has. Therefore, we can set up the equation:

\[
3.8t = 3t + 0.2
\]

### Solving the equation:
1. Subtract \( 3t \) from both sides to isolate the terms involving \( t \):
   \[
   3.8t - 3t = 0.2
   \]
   \[
   0.8t = 0.2
   \]
2. Divide by 0.8 to solve for \( t \):
   \[
   t = \frac{0.2}{0.8} = 0.25 \text{ hours}
   \]

### Conversion to minutes:
To convert 0.25 hours to minutes:
\[
0.25 \times 60 = 15 \text{ minutes}
\]

So, it will take Wyatt **15 minutes** to catch up to Kirk.

Would you like any further explanation or have other questions?

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Here are 5 related questions to consider:

1. How far will both swimmers have traveled when Wyatt catches up to Kirk?
2. If Wyatt increased his speed by 0.5 km/h, how much time would he save in catching up to Kirk?
3. What if Kirk increased his speed to 3.5 km/h—how long would it take for Wyatt to catch up then?
4. If Wyatt had a 0.1 km head start instead, how would that change the time needed for Kirk to catch up?
5. How would the time change if both swimmers had to swim for an additional 2 km after Wyatt catches up?

**Tip:** When dealing with relative speed problems, focus on the difference in speeds and the distance separating the two objects or people to set up the right equation.

Discover how long it will take Wyatt to catch up to Kirk while swimming across Crystal Lake. This problem involves calculating relative speeds, distances covered, and solving equations with variables. Suitable for Grades 7-9.

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