Esmerelda rents a car from a company that rents cars by the hour. She has to pay an initial fee of $50

, and then they charge her $9
per hour. She has $200
available to spend on car rental. What is the greatest number of hours for which she can rent the car? (The car cannot be rented for part of an hour.)

1 answer

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.

  1. Initial fee: $50
  2. Hourly charge: $9 per hour
  3. 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.