Answer :
To understand which graph correctly represents the charge for a cab ride of [tex]\( x \)[/tex] miles, let's break down the charges based on the described rates.
1. Flag Drop Fee (0 miles):
- For a ride of 0 miles, the charge is \[tex]$5.00. So, when \( x = 0 \), \( y = 5.0 \). 2. For \( 0 \leq x < 10 \) miles: - Each mile from 0 to less than 10 miles costs \$[/tex]1.00 per mile in addition to the \[tex]$5.00 flag drop fee. - Therefore, if \( x \) is the number of miles, the formula for the total charge \( y \) is: \[ y = 5.0 + 1.0 \cdot x \] 3. For \( x \geq 10 \) miles: - The first 10 miles are charged at \$[/tex]1.00 per mile and cost a total of \[tex]$10.00 plus the initial \$[/tex]5.00 flag drop fee, making it \[tex]$15.00. - Any miles beyond 10 are charged at \$[/tex]2.00 per mile.
- If [tex]\( x \)[/tex] is the number of miles where [tex]\( x \geq 10 \)[/tex], the formula for the total charge [tex]\( y \)[/tex] is:
[tex]\[ y = 15.0 + 2.0 \cdot (x - 10) \][/tex]
Let us compile the charges for a sample of miles:
[tex]\[ \begin{array}{|c|c|} \hline \text{Miles} & \text{Charge (\$)} \\ \hline 0 & 5.0 \\ 1 & 6.0 \\ 2 & 7.0 \\ 3 & 8.0 \\ 4 & 9.0 \\ 5 & 10.0 \\ 6 & 11.0 \\ 7 & 12.0 \\ 8 & 13.0 \\ 9 & 14.0 \\ 10 & 15.0 \\ 11 & 17.0 \\ 12 & 19.0 \\ 13 & 21.0 \\ 14 & 23.0 \\ 15 & 25.0 \\ 16 & 27.0 \\ 17 & 29.0 \\ 18 & 31.0 \\ 19 & 33.0 \\ 20 & 35.0 \\ \hline \end{array} \][/tex]
With these values:
- The charge starts from \[tex]$5.0 at \( x = 0 \). - From \( x = 1 \) to \( x = 9 \), the charge increases linearly by $[/tex]1.00 for each mile.
- At [tex]\( x = 10 \)[/tex], the charge is \[tex]$15.0. - For \( x > 10 \), the charge increases by $[/tex]2.00 per mile starting from \$15.0 at [tex]\( x = 10 \)[/tex].
With these insights, the graph should have:
- An initial point at [tex]\( (0, 5.0) \)[/tex].
- A linear increase with a slope of 1 from [tex]\( x = 0 \)[/tex] to [tex]\( x = 10 \)[/tex].
- A linear increase with a slope of 2 starting from [tex]\( (10, 15.0) \)[/tex].
Therefore, the correct graph will show a piecewise linear function with the above-described properties.
1. Flag Drop Fee (0 miles):
- For a ride of 0 miles, the charge is \[tex]$5.00. So, when \( x = 0 \), \( y = 5.0 \). 2. For \( 0 \leq x < 10 \) miles: - Each mile from 0 to less than 10 miles costs \$[/tex]1.00 per mile in addition to the \[tex]$5.00 flag drop fee. - Therefore, if \( x \) is the number of miles, the formula for the total charge \( y \) is: \[ y = 5.0 + 1.0 \cdot x \] 3. For \( x \geq 10 \) miles: - The first 10 miles are charged at \$[/tex]1.00 per mile and cost a total of \[tex]$10.00 plus the initial \$[/tex]5.00 flag drop fee, making it \[tex]$15.00. - Any miles beyond 10 are charged at \$[/tex]2.00 per mile.
- If [tex]\( x \)[/tex] is the number of miles where [tex]\( x \geq 10 \)[/tex], the formula for the total charge [tex]\( y \)[/tex] is:
[tex]\[ y = 15.0 + 2.0 \cdot (x - 10) \][/tex]
Let us compile the charges for a sample of miles:
[tex]\[ \begin{array}{|c|c|} \hline \text{Miles} & \text{Charge (\$)} \\ \hline 0 & 5.0 \\ 1 & 6.0 \\ 2 & 7.0 \\ 3 & 8.0 \\ 4 & 9.0 \\ 5 & 10.0 \\ 6 & 11.0 \\ 7 & 12.0 \\ 8 & 13.0 \\ 9 & 14.0 \\ 10 & 15.0 \\ 11 & 17.0 \\ 12 & 19.0 \\ 13 & 21.0 \\ 14 & 23.0 \\ 15 & 25.0 \\ 16 & 27.0 \\ 17 & 29.0 \\ 18 & 31.0 \\ 19 & 33.0 \\ 20 & 35.0 \\ \hline \end{array} \][/tex]
With these values:
- The charge starts from \[tex]$5.0 at \( x = 0 \). - From \( x = 1 \) to \( x = 9 \), the charge increases linearly by $[/tex]1.00 for each mile.
- At [tex]\( x = 10 \)[/tex], the charge is \[tex]$15.0. - For \( x > 10 \), the charge increases by $[/tex]2.00 per mile starting from \$15.0 at [tex]\( x = 10 \)[/tex].
With these insights, the graph should have:
- An initial point at [tex]\( (0, 5.0) \)[/tex].
- A linear increase with a slope of 1 from [tex]\( x = 0 \)[/tex] to [tex]\( x = 10 \)[/tex].
- A linear increase with a slope of 2 starting from [tex]\( (10, 15.0) \)[/tex].
Therefore, the correct graph will show a piecewise linear function with the above-described properties.
