Question

Three hourly rental car companies I, II, and III charge customers based on the time duration for which they book a car along with the fixed basic pay for booking for a day.
Company I:


Company II:
$\begin{array}{cc}\text{Travel time}& \text{Booking price}\left(\$\right),y\\ \text{(in hr), x}& \\ 0& 10\\ 3& 31\\ 5& 45\\ 7& 59\end{array}$

Company III: Charges fixed pay of $\$12$ and its hourly price is an integer between hourly prices of company I and company II.
If you want to book for 6 hours, which rental car company would you choose?
Assume you will choose the company which is charging you the lowest price of booking.

• A. Company II
• B. Either Company I or III
• C. Company III
• D. Company I

Answer 1

The correct option is C Company III

Detailed step-by-step solution:
Finding the equation of cab service company I:
y-intercept: It is a point where the graph of the line touches or crosses the y-axis. The x-coordinate is zero at this point.
The graph touches the y-axis at $(0,20).$
$\Rightarrow$ y-intercept, $b=20$
Consider two points $(0,20)$ and $(4,40)$ on the graph:
$\text{Slope, m}=\frac{\text{Rise}}{\text{Run}}$
$=\frac{\text{Change in y}}{\text{Change in y}}$
$=\frac{40-20}{4-0}$
$=\frac{20}{4}$
$=5$
Unit rate $=$ Slope $\left(m\right)=\$5$ per hour (hourly price of booking)
The equation of the straight line is given by $y=mx+b.$
At $x=0,y=20.$
So, the equation will be $y=5x+20.$
Total booking price for 6 hours from hourly rental car company I:
Total booking price for 6 hours from hourly rental car company I will be the value of $y$ when $x=6.$
$\Rightarrow y=5\left(6\right)+20$
$\Rightarrow y=30+20$
$\Rightarrow y=50$
Total booking price for $6$ hours from hourly rental car company $I=\$50$.
Finding the equation of hourly rental car company II:
y-intercept is the value of $y$ when $x=0.$
From the table, at $x=0,y=10.$
$\Rightarrow$ y-intercept, $b=10$
Consider two ordered pairs $(0,10)$ and $(3,31)$ on the graph.
$\text{Slope, m}=\frac{\text{Rise}}{\text{Run}}$
$=\frac{\text{Change in y}}{\text{Change in y}}$
$=\frac{31-10}{3-0}$
$=\frac{21}{3}$
$=7$
Unit rate $=$ Slope $\left(m\right)=\$7$ per hour (hourly price of booking)
The equation of the straight line is given by $y=mx+b.$
The equation will be $y=7x+10.$
Total booking price for 6 hours from hourly rental car company II:
Total booking price for 6 hours from hourly rental car company II will be the value of y when $x=6.$
$\Rightarrow y=7\left(6\right)+10$
$\Rightarrow y=42+10$
$\Rightarrow y=52$
Total booking price for 6 hours from hourly rental car company II $=\$52$
Finding the equation of hourly rental car company III:
Let the equation be $y=mx+b,$ where $x$ represents the travel time (in hr) and $y$ represents the booking price $\left(\$\right).$
y-intercept is the value of y when $x=0.$
Fixed price $=\$12$
I.e., at $x=0,y=\$12$
$\Rightarrow$ y-intercept, $b=12$
Hourly price of company III is an integer between hourly prices of company I and company II.
Hourly prices of company $I=\$5$ per hour
Hourly prices of company $II=\$7$ per hour
The integer that lies between $5$ and $7$ is $6.$
=> Hourly prices of company $III=\$6$ per hour
Hourly prices of company $III=$ Slope of the equation
$\Rightarrow$ Slope $\left(m\right)=\$6$ per hour
The equation will be $y=6x+12.$
Total booking price for $6$ hours from hourly rental car company III:
Total booking price for $6$ hours from hourly rental car company III will be the value of y when $x=6.$
$\Rightarrow y=6\left(6\right)+12$
$\Rightarrow y=36+12$
$\Rightarrow y=48$
Total booking price for $6$ hours from hourly rental car company III $=\$48$
Out of the three companies, company III has the lowest booking price for 6 hours, i.e., $\$48.$
Hence, you would like to choose hourly rental car company III for booking.
$\Rightarrow$ Option C is correct.