To determine the greatest number of hours Esmerelda can rent the car, we need to set up the equation based on her available budget and the costs involved.
- Initial fee: $50
- Hourly charge: $9 per hour
- Total budget: $200
The total cost for renting the car can be expressed as:
\[ \text{Total Cost} = \text{Initial fee} + (\text{Hourly charge} \times \text{Number of hours}) \] Substituting the known values into the equation, we have:
\[ \text{Total Cost} = 50 + 9h \]
where \(h\) is the number of hours she rents the car.
Since she has $200 available, we set up the inequality:
\[ 50 + 9h \leq 200 \]
To find \(h\), we first subtract $50 from both sides:
\[ 9h \leq 200 - 50 \] \[ 9h \leq 150 \]
Now, we divide both sides by 9:
\[ h \leq \frac{150}{9} \]
Calculating the right side gives:
\[ h \leq 16.6667 \]
Since Esmerelda cannot rent the car for a fraction of an hour, we take the greatest whole number less than or equal to 16.6667:
\[ h \leq 16 \]
Thus, the greatest number of hours for which she can rent the car is:
\[ \boxed{16} \]
To verify, let’s calculate the total cost if she rents the car for 16 hours:
\[ \text{Total Cost} = 50 + 9 \times 16 = 50 + 144 = 194 \]
She would spend $194, which is within her budget of $200. If she rented the car for 17 hours:
\[ \text{Total Cost} = 50 + 9 \times 17 = 50 + 153 = 203 \]
This exceeds her budget of $200. Therefore, the confirmed maximum rental time is indeed:
\[ \boxed{16} \] hours.