For Rip-Off rentals, there are two separate variables determining the price.
B
b
Substitute d=2 and m=10, 20, and 100 into the functions from Part A.
A
a
Rip-Off rental: C=25d+0.5m
Teacher: C=0.03(2)^(m-1)
B
b
Under 10 miles: Teacher's offer is better.
20 miles or over: Rip-Off Rentals is better.
Practice makes perfect
a
Let's call the rental cost C. If we rent the car from Rip-Off Rentals, we are charged $25 per day and 50¢ per mile. If the number of days the car is rented is d, and the number of miles driven is m, we can write the following equation to describe the cost when using Rip-Off Rentals.
C=25d+0.5mThe amount your teacher charges only depends on the number of miles you drive, m. We can describe it with a geometric sequence.
With this information, we can write the function that the teacher uses.
C=0.03(2)^(m-1)
Note that both functions are only defined for whole numbers, which means our functions are step functions.
b
To determine which is the better deal, we should substitute d=2 and m=10, 20, and 100 into the functions and compare the costs.
Only the function from Rip-Off rentals includes the number of days d. Since we are only considering a 2 day rental, we can substitute d= 2 into the function for Rip-Off rental.
C=25( 2)+0.5m ⇔ C=50+0.5mNow we can calculate the costs of the different deals. Let's start with Rip-Off Rentals.
c|c|c
m & 50+0.5m & C
10 & 50+0.5( 10) & 55
20 & 50+0.5( 20) & 60
100 & 50+0.5( 100) & 100 [0.3em]
Next, we will calculate the rental fee that your teacher charges.
c|c|c
m & 0.03(2)^(m-1)& C
10 & 0.03(2)^(10-1) & 15.56
20 & 0.03(2)^(20-1) & 15 728
100 & 0.03(2)^(100-1) & 1.9(10)^(28)
As we can see, Rip-Off Rentals is more expensive if you plan to drive the car for 10 miles or less. However, as the number of miles increase, the teacher's offer becomes very expensive. This is because the teacher uses an exponential model to calculate the price.
