Question

9. You need to rent a car and compare the charges of threedifferent companies. Company A charges 20 cents permile plus $20 per day. Company B charges 10 cents permile plus $35 per day. Company C charges $70 per daywith no mileage charge.(a) Find formulas for the cost of driving cars rentedfrom companies A, B, and C, in terms of x, thedistance driven in miles in one day.(b) Graph the costs for each company for 0 ≤ x ≤ 500.Put all three graphs on the same set of axes.(c) What do the slope and the vertical intercept tell youin this situation?(d) Use the graph in part (b) to find under what circum-stances company A is the cheapest. What aboutCompany B? Company C? Explain why your re-sults make sense.

Answer 1

As required, we will start from the part b. First we will write functions for the cost of renting a car in each companies in terms of the distance x in miles

`c_A(x)=0.20x\text{ +20}`

`c_B(x)=0.10x+35\text{ }`
`c_C(x)=70`

Part b:

We have to graph the three function in the same axes, in the range 0 < x< 500

To do thus, let us see the graph and then we will analyze it

Now, we are going to analyze this graph, the red line is the cost of the company c, the green line is the cost of company B, and the purple line is the cost for the company A

As all the three curves are straight lines, to graph them we only must know two points. We will determine a point in the next part, so we only need more one, one posibility is to find the cost for each company when x=1, that is, when is covered one mile.

Part c;

we will find the intercept of each curve with the y axe

`c_A(0)=0.20*0+20\text{ the point in the graph is \lparen0,20\rparen}`
`c_B(0)\text{ =35 the point in the graph is \lparen0,35\rparen}`
`c_C(0)\text{ =70 The point in the graph is \lparen0,70\rparen}`

we can see this starting cost in the next figure:

We will find the slope of the three lines

For this, we will compare the equation of the functions with the general equation

`y=mx+b`

here we have that m represent the slope and b the intercept , therefore, we have the next conclusions

* For the company A, the slope is 20;

* For the company B the slope is 10

* For the company C, the slope is 0,

Part d:

As the slope can be interpreted as the rate of growth of a function, in this case the rate of growth of the cost in renting a car, we can say that:

The company has the greater cost of all the companies, however if the car will be used to travel longer distances (about 350 miles or more) is better to rent in the company C than rent in the other two companies.

For medium distances ( between 150 miles and 350 miles) the best option is the company B,

For short distances (at most 150 ,miles), the best company is the company A, But as the cost of rent growth faster than the other two companies, rapidly A left to be the best option