Math Is Fun Forum

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Car Rental Costs

An economy car rented in Florida from National Car Rental on a weekly basis costs $95 per week. Extra days cost $24 per day until the day rate exceeds the weekly rate, in which case the weekly rate applies. Also, any part of a day used counts as a full day. Find the cost C of renting an economy car as a function of the number x of days used, where 7 is less than or equal to x less than or equal to 14. Graph the function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Car Rental Costs

Find out, first, how many days will cause the daily rate charge to exceed the weekly charge. That fixes when the rate change occurs.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Car Rental Costs

Find out, first, how many days will cause the daily rate charge to exceed the weekly charge. That fixes when the rate change occurs.

Bob

Can this be solved without creating a function?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Car Rental Costs

It says find the cost as a function then later graph the function.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Car Rental Costs

It says find the cost as a function then later graph the function.

Bob

Ok. Let me see what I can do.

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Car Rental Costs

It says find the cost as a function then later graph the function.

Bob

Let me see.

From the problem, I know that:

-per week probably means 7 days in a week.
- Renting on a weekly basis costs $95 per week.
- Extra days (beyond the weekly rate) cost $24 per day until the day rate exceeds the weekly rate.
- Any part of the day used counts as a full day.

Based on this information, I came up with the following piecewise function to represent the cost:

For 7 ≤ x ≤ 14:

C(x) = 95  if x = 7...top portion

C(x) = 95 + (x - 7)( 24)   if 8 ≤ x ≤ 14...bottom portion

Is this right, Bob?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Car Rental Costs

I don't think so. Here's what I have done.

x                                                          cost
7                                                           95
8                                                           95 + 1x24
9                                                           95 + 2x24
10                                                         95 + 3x24
11                                                         95 + 95
12                                                         95
13                                                         95
14                                                         95

You get charged for the first week at 95. After that you have to add 24 for each extra day until you get to 11 days.  At that point 95 + 4x24 would be more than the weekly charge so that applies instead. Your formula is right up to and including all of day 10. Thereafter the two week charge applies. I wrote this as 10< x ≤ 14, not 11≤ x ≤ 14 because this makes clear what happens if you only rent for one hour on day eleven.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Car Rental Costs

I don't think so. Here's what I have done.

x                                                          cost
7                                                           95
8                                                           95 + 1x24
9                                                           95 + 2x24
10                                                         95 + 3x24
11                                                         95 + 95
12                                                         95
13                                                         95
14                                                         95

You get charged for the first week at 95. After that you have to add 24 for each extra day until you get to 11 days.  At that point 95 + 4x24 would be more than the weekly charge so that applies instead. Your formula is right up to and including all of day 10. Thereafter the two week charge applies. I wrote this as 10< x ≤ 14, not 11≤ x ≤ 14 because this makes clear what happens if you only rent for one hour on day eleven.

Bob

Just to be clear, can you please provide the correct piecewise function? I would like to move on from this application.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Car Rental Costs

I have

7  ≤ x  ≤ 10  C = 95 + 24(x-7)

10 < x ≤ 14   C = 190

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Car Rental Costs

I have

7  ≤ x  ≤ 10  C = 95 + 24(x-7)

10 < x ≤ 14   C = 190

Bob

We are done here, right?