Question: Solving a Rate Problem for Distance
In training for a triathlon, Yuna swam 1.9 miles in the open ocean, which took her 2 hours. For the first part of her swim, she averaged 1.4 miles per hour. For the second part of her swim, she swam against the current and started to tire, so her average speed decreased to 0.8 mph.
What was the distance of the first part of Yuna’s swim? Round to the nearest tenth as needed.
The distance in the first part of her swim was 0.4 miles.
Rate (mi/h) Time (h) Distance (mi)
Part 1 1.4 t 1.4t
Part 2 0.8 2 - t 0.8(2 - t)
1. **State the problem:**
Yuna swims a total distance of $1.9$ miles in $2$ hours. She swims the first part at $1.4$ mph for $t$ hours and the second part at $0.8$ mph for $2 - t$ hours. We need to find the distance of the first part of her swim.
2. **Set up the equation:**
The total distance is the sum of the distances of the two parts:
$$1.4t + 0.8(2 - t) = 1.9$$
3. **Distribute and simplify:**
$$1.4t + 1.6 - 0.8t = 1.9$$
Combine like terms:
$$ (1.4t - 0.8t) + 1.6 = 1.9$$
$$0.6t + 1.6 = 1.9$$
4. **Isolate $t$:**
Subtract $1.6$ from both sides:
$$0.6t + \cancel{1.6} - \cancel{1.6} = 1.9 - 1.6$$
$$0.6t = 0.3$$
Divide both sides by $0.6$:
$$\frac{0.6t}{\cancel{0.6}} = \frac{0.3}{\cancel{0.6}}$$
$$t = 0.5$$
5. **Find the distance of the first part:**
Distance $= 1.4 \, \times \, t = 1.4 \, \times \, 0.5 = 0.7$ miles.
6. **Check the problem statement:**
The problem states the distance was $0.4$ miles, but our calculation shows $0.7$ miles based on the given rates and total distance.
7. **Conclusion:**
The distance of the first part of Yuna's swim is approximately **$0.7$ miles**.
**Note:** The problem's given answer $0.4$ miles does not match the calculation from the data provided.