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In Delray Beach, Florida, you can take a luxury golf cart ride around downtown. The driver charges $4 for the first 1/4 mile, plus $1.50 for each additional 1/2 mile. Which inequality represents the number of miles, \( m \), that you could ride and pay no more than $10?

A. \( 3.25 + 1.5m \leq 10 \)

B. \( 3.25 + 3m \leq 10 \)

C. \( 4 + 1.5m \leq 10 \)

D. \( 4 + 3m \leq 10 \)

Answer :

To solve this problem, we need to understand the costs involved in the luxury golf cart ride around downtown Delray Beach, Florida, and express them as an inequality representing the number of miles, [tex]m[/tex], that you could ride while paying no more than $10.

Let's break it down step-by-step:

  1. Initial Cost: The first 1/4 mile costs $4.

    This is a fixed cost and applies to every ride regardless of the distance traveled beyond 1/4 mile.

  2. Additional Cost: Each additional 1/2 mile costs $1.50.

    If you travel more than 1/4 mile, this charge applies to the extra distance.

  3. Setting Up the Inequality: You want to spend no more than $10 for the ride.

    • The total cost can be expressed as:

      [tex]\text{Total Cost} = 4 + 1.5 \times \text{Number of additional half-miles}[/tex]

    • We need to account for the distance in miles, [tex]m[/tex]. Since the first 1/4 mile is already covered with the initial $4, the additional distance beyond the first 1/4 mile is [tex]m - \frac{1}{4}[/tex].

    • Convert [tex]m - \frac{1}{4}[/tex] miles into half-mile segments since each additional 1/2 mile costs $1.50. Hence, the number of additional half-miles is [tex]\frac{m - \frac{1}{4}}{\frac{1}{2}} = 2(m - \frac{1}{4})[/tex].

    • Substitute this into the cost equation:

      [tex]\text{Total Cost} = 4 + 1.5 \times 2(m - \frac{1}{4})[/tex]

    • Simplifying this gives:

      [tex]\text{Total Cost} = 4 + 3(m - \frac{1}{4})[/tex]

    • Continue simplifying:

      [tex]\text{Total Cost} = 4 + 3m - \frac{3}{4}[/tex]

    • Simplify further:

      [tex]\text{Total Cost} = 3m + 3.25[/tex]

  4. Setting Up the Inequality: To ensure the total cost doesn’t exceed $10:

    [tex]3m + 3.25 \leq 10[/tex]

  5. Solving the Inequality:

    • Subtract 3.25 from both sides:

      [tex]3m \leq 10 - 3.25[/tex]

      [tex]3m \leq 6.75[/tex]

    • Divide both sides by 3 to solve for [tex]m[/tex]:

      [tex]m \leq \frac{6.75}{3}[/tex]

      [tex]m \leq 2.25[/tex]

Thus, the correct inequality representing the situation is option D: 4 + 3m \leq 10. You can ride up to [tex]2.25[/tex] miles while keeping the total cost no more than $10.

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