Question

piece - wise defined functions
1.13a
homeworksheet
write a piecewise defined function for each problem below and the answer the questions

1. to lease a car is to essentially rent the car for a period of time. you lease a car for four years with a mileage limit. this limit means that the cost to lease the car over four years depends on the number of miles you drive during that time.

the cost of the lease for 4 years is $12,000 if you drive any amount up to 42,000 miles. for each additional mile over 42,000 you will be charged $0.25 per mile up to 52000 miles. for each additional mile driven farther than 52,000 you will be charged at a rate of $0.50 per mile. write a piece wise defined function c(m) for the cost, c, of your car’s lease based on the number of miles driven, m.

Explanation:

Step1: Identify the intervals

We have three intervals for the number of miles \( m \):

• \( 0 \leq m \leq 42000 \): In this interval, the cost is a flat rate of $12,000. So the function is \( C(m)=12000 \).
• \( 42000 < m \leq 52000 \): Here, we first pay the flat rate of $12,000 and then an additional $0.25 per mile over 42,000. The number of additional miles is \( m - 42000 \). So the cost function is \( C(m)=12000 + 0.25(m - 42000) \). Simplifying this: \( C(m)=12000 + 0.25m - 10500 = 0.25m + 1500 \).
• \( m > 52000 \): Now, we pay the flat rate, the additional cost up to 52,000 miles, and then an additional $0.50 per mile over 52,000. First, calculate the cost up to 52,000 miles using the previous interval's function: \( C(52000)=0.25(52000)+1500 = 13000 + 1500 = 14500 \). The number of miles over 52,000 is \( m - 52000 \). So the cost function is \( C(m)=14500 + 0.50(m - 52000) \). Simplifying: \( C(m)=14500 + 0.50m - 26000 = 0.50m - 11500 \).

Step2: Write the piece - wise function

Combining these three cases, the piece - wise defined function for the cost \( C(m) \) of the car's lease based on the number of miles driven \( m \) is:

$$ C(m)=
LATEXBLOCK0
$$

Answer:

The piece - wise function is \(\boldsymbol{C(m)=

$$\begin{cases}12000, & 0\leq m\leq42000\\0.25m + 1500, & 42000 < m\leq52000\\0.50m - 11500, & m > 52000\end{cases}$$

}\)