To find the distance Yuna swam during the first part of her swim, we need to solve for the variable \( t \), which represents the time she spent swimming during that first part.
Step-by-Step Solution:
1. Identify the given information:
- Total distance of the swim: \( 1.9 \) miles
- Total time of the swim: \( 2 \) hours
- Speed during the first part: \( 1.4 \) miles per hour
- Speed during the second part: \( 0.8 \) miles per hour
2. Set up the expressions for the distances in each part of the swim:
- Let \( t \) be the time in hours for the first part of the swim.
- The distance for the first part is \( 1.4t \) miles.
- The remaining time for the second part of the swim is \( 2 - t \) hours.
- The distance for the second part is \( 0.8(2 - t) \) miles.
3. Write the equation for the total distance:
[tex]\[
1.4t + 0.8(2 - t) = 1.9
\][/tex]
4. Simplify and solve the equation:
[tex]\[
1.4t + 0.8(2 - t) = 1.9
\][/tex]
Distribute the \( 0.8 \):
[tex]\[
1.4t + 1.6 - 0.8t = 1.9
\][/tex]
Combine like terms:
[tex]\[
(1.4t - 0.8t) + 1.6 = 1.9
\][/tex]
[tex]\[
0.6t + 1.6 = 1.9
\][/tex]
Subtract \( 1.6 \) from both sides:
[tex]\[
0.6t = 0.3
\][/tex]
Divide by \( 0.6 \):
[tex]\[
t = \frac{0.3}{0.6} = 0.5
\][/tex]
5. Calculate the distance of the first part of the swim:
The distance for the first part is:
[tex]\[
1.4 \times 0.5 = 0.7 \text{ miles}
\][/tex]
Therefore, the distance of the first part of Yuna's swim was [tex]\( 0.7 \)[/tex] miles.
