To determine the speed at which Marina runs, let's break down the problem step-by-step.
First, we need to understand the relationships between distance, rate (speed), and time. The formula we use is:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Rate}} \][/tex]
Given the problem's details, let's define the variables and set up the corresponding equations:
1. Define the variables:
- Let [tex]\( r \)[/tex] be the speed at which Marina runs in miles per hour (mph).
2. Set up for bicycling:
- Distance: [tex]\( 19.5 \)[/tex] miles
- Rate: [tex]\( r + 9 \)[/tex] mph (since she bikes 9 mph faster than she runs)
- Time biking: [tex]\( \frac{19.5}{r + 9} \)[/tex]
3. Set up for running:
- Distance: [tex]\( 6 \)[/tex] miles
- Rate: [tex]\( r \)[/tex] mph
- Time running: [tex]\( \frac{6}{r} \)[/tex]
4. Determine the relationship between the two times:
The problem states that the time taken to bicycle 19.5 miles is the same as the time taken to run 6 miles. Therefore, we set the two time equations equal to each other:
[tex]\[
\frac{19.5}{r + 9} = \frac{6}{r}
\][/tex]
5. Solve for [tex]\( r \)[/tex]:
Cross-multiply to eliminate the fractions:
[tex]\[
19.5r = 6(r + 9)
\][/tex]
Distribute [tex]\( 6 \)[/tex] on the right side:
[tex]\[
19.5r = 6r + 54
\][/tex]
Move [tex]\( 6r \)[/tex] to the left side by subtracting [tex]\( 6r \)[/tex] from both sides:
[tex]\[
19.5r - 6r = 54
\][/tex]
Simplify:
[tex]\[
13.5r = 54
\][/tex]
Divide both sides by 13.5 to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{54}{13.5}
\][/tex]
Simplify:
[tex]\[
r = 4
\][/tex]
So, the speed at which Marina runs is [tex]\( \boxed{4} \)[/tex] miles per hour.
