To determine at what speed Marina runs, let's analyze the given problem step-by-step:
1. Establish Variables:
- Let [tex]\( r \)[/tex] be Marina's running speed in miles per hour (mph).
- Therefore, Marina's bicycling speed would be [tex]\( r + 9 \)[/tex] mph, since she bikes 9 mph faster than she runs.
2. Time Calculation:
- Time is calculated as distance divided by speed.
- Time spent bicycling is given by:
[tex]\[
\text{Time for bicycling} = \frac{19.5 \text{ miles}}{r + 9 \text{ mph}}
\][/tex]
- Time spent running is given by:
[tex]\[
\text{Time for running} = \frac{6 \text{ miles}}{r \text{ mph}}
\][/tex]
3. Equalizing the Times:
- Marina's time spent bicycling is equal to the time spent running:
[tex]\[
\frac{19.5}{r + 9} = \frac{6}{r}
\][/tex]
4. Solving for [tex]\( r \)[/tex]:
- To solve this equation, we cross-multiply:
[tex]\[
19.5 \cdot r = 6 \cdot (r + 9)
\][/tex]
- Simplify the equation:
[tex]\[
19.5r = 6r + 54
\][/tex]
[tex]\[
19.5r - 6r = 54
\][/tex]
[tex]\[
13.5r = 54
\][/tex]
[tex]\[
r = \frac{54}{13.5}
\][/tex]
[tex]\[
r = 4
\][/tex]
Thus, Marina runs at a speed of 4 miles per hour.
Given the options:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
4 mph & 5 mph & 9 mph & 13 mph \\
\hline
\end{tabular}
\][/tex]
The correct answer is [tex]\( 4 \)[/tex] mph.
