A rental car company offers two rental plans, Plan A and Plan B, for the same economy size car. For both plans, the total rental cost is a function of the number of miles that the car is driven. In addition to a flat fee of $75, Plan A offers a rate of $0.20 per mile for an unlimited number of miles. Plan B offers a higher mileage rate of $0.35 per mile but does not charge a flat fee for the rental. Create a system of linear functions modeling the cost of the car rental plans, A and B, as a function of the miles driven.
For how many miles will the rental fee be the same under both plans A and B?

a. C(x) = 75 + 0.2x and C'(x) = $0.35x b. 500 miles

Step-by-step explanation:

a. Create a system of linear functions modeling the cost of the car rental plans, A and B, as a function of the miles driven.

Let x be the number of miles driven by the car for each plan.

For plan A, since there is a $75 initial flat fee and a charge of $0.20 per mile, after moving x miles, the extra cost is $0.20x. So, the total cost C for plan A is

C(x) = 75 + 0.2x

For plan B since there is no initial flat fee but only a charge of $0.35 per mile, for x miles, the cost for plan B is C'(x) = $0.35x

b. For how many miles will the rental fee be the same under both plans A and B?

When the rental fees are the same, we equate both expressions.

So, 75 + 0.2x = 0.35x

collecting like terms, we have

75 = 0.35x - 0.2x

75 = 0.15x

dividing both sides by 0.15, we have

x = 75/0.15

= 500 miles

So, the rental fee will be the same after 500 miles

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for