Q40. - Algebra 2

Question

For Exercises 39-41, use the following information.

Adam and his family are planning to rent a midsize car for one-day trip. In the Standard Rental Plan, they can rent a car for \(52 per day plus 23 cents per mile. In the Deluxe Rental Plan, they can rent a car for \)80 per day with unlimited mileage. 

40. Graph these equations. Estimate the break-even point of the rental costs.

Step-by-Step Solution

Answer

The graph of the equations $y=0.23x+52$ and $y=80$ is provided below. The break-even point is $\left(121.7,80\right)$.

Step-1 – Interpret the data

According to the question, there are two rental plans.

The cost of renting a car as per Standard Rental plan is $52 per day plus 23 cents per mile. In the Deluxe Rental Plan, the cost to rent a car is $80 per day with unlimited mileage. 

Step-2 – Define the variables

Consider two variables x and y.

Denote the cost of renting a car by y and number of miles driven by x.

Step-3 – Build the equations

Let the number of miles driven be x.

In Standard Rental plan car is rented for $52 per day plus 23 cents per mile. Therefore, the cost to rent a car is expressed below. The per day cost is fixed as one-day trip is to be scheduled.

$y=0.23x+52$

In Deluxe Rental plan car is rented for $80 per day with unlimited mileage. 

Therefore, the cost to rent a car is expressed below. 

$y=80$

 

Thus, the equation that represent cost of renting a car as per Standard Rental plan is $y=0.23x+52$ and cost of renting a car as per Deluxe Rental plan is $y=80$.

Step-4 – Apply the concept of slope-intercept form

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step-5 –Write the equations in slope-intercept form

Consider the first equation $y=0.23x+52$.

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $0.23$ and intercept of y-axis c is $52$.

Now, consider the second equation $y=80$.

Rewrite the equation in form of slope-intercept form.

$y=0x+80$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is 0 and intercept of y-axis c is 80.

Step-6 – Identify the point of intersection of the equations

Plot the equations on the same plane and the point where both the equations intersect is the break-even point of the system of the equations.

The red line denotes the equation $y=0.23x+52$ and blue line denotes the equation $y=80$.

Therefore, the break-even point of the rental costs is $\left(121.7,80\right)$.