Story problem help please

Question

There are two car rental options available for a five-day trip. Option I is $50 per day, with 200 free miles and $0.40 per mile for each additional mile. Option II is $65.28 per day, with a charge of $0.25 per mile.

 

(a) Determine the cost C of a 600-mile trip for both options. (Round all charges to the nearest dollar.)
Option I C₁ = $???? 

Option II C₂ = $????


(b) Model the data with a cost function for each five-day option. (Enter all functions exactly. Simplify your answer completely.)

 

 

          =???? if 0 ≤ x ≤ 200 

C₁(x) = ??? if x >200

 

 

C₂(x)=??? if x≥0
if
0

b) Determine the mileages at which each option is preferable. (Round all distances to the nearest mile.)

 

Option I is preferable if x ∈ [0,???)

Option II is preferable if x> ???

 

Answer 1

Alright, so this can be pretty confusing because it looks like we'll need to utilize piecewise functions to create correct equation. Also, I'll be using m to represent miles.

 

C₁ = 250 if m<= 200

 and 250 + .4(m - 200) if m > 200

 

C₂ = 326.40 + .25m

 

So on a)

 

We plug 600 in for m and use the second equation on C₁ because m is greater than 200.

 

C₁ = 250 + .4(600-200)

 

     = 250 + .4(400)

 

     = 250 + 160

 

     = $410

 

And,

 

C₂ = 326.40 + .25(600)

 

     = 326.40 + 150

 

     = $476.40

 

And on b)

 

There will be a point where these equations cross each other and the second option becomes more economical, the best way to picture is as the place where they meet (or where the two equations are equal to each other)

 

So, 250 + .4(m-200) = 326.40 + .25m

 

250 + .4m - 80 = 326.40+ .25m

 

170 +.4m = 326.40 +.25m

 

170 +.15m = 326.40

 

.15m = 156.40

 

m = 1042.666667 or about 1043 miles.  

 

So C₁ is best if m<1043

 

And C₂ is best if m>=1043