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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.

\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & \begin{tabular}{c}
Rate \\
[tex]$(mi/h)$[/tex]
\end{tabular} & \begin{tabular}{c}
Time \\
[tex]$(h)$[/tex]
\end{tabular} & \begin{tabular}{c}
Distance \\
[tex]$(mi)$[/tex]
\end{tabular} \\
\hline
Part 1 & 1.4 & [tex]$t$[/tex] & [tex]$1.4t$[/tex] \\
\hline
Part 2 & 0.8 & [tex]$2-t$[/tex] & [tex]$0.8(2-t)$[/tex] \\
\hline
\end{tabular}

What was the distance of the first part of Yuna's swim? Round to the nearest tenth as needed.

The distance of the first part of her swim was [tex]$\square$[/tex] miles.



Answer :

GinnyAnswer
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.