A rental car company charges a base fee of $40 plus $0.25 per mile for the first 100 miles the car is driven. The company charges the same base fee plus a reduced price of $0.18 per mile for cars driven over 100 miles. The piecewise function below represents the different amounts the company charges.
[tex]y=\left\{\begin{array}{l}40+0.25 m, m \leq 100 \\ 40+0.18 m, m\ \textgreater \ 100\end{array}\right.[/tex]
If someone drives their rental car 150 miles, how much will they owe the rental company?
A. $37.50
B. $77.50
C. $67.00
D. $27.00
Answer (2)
A student driving 150 miles will owe the rental car company $67.00. This is calculated using the appropriate equation from the piecewise function for distances greater than 100 miles. Therefore, the correct choice is C. $67.00.
;
Determine that since 150 miles > 100 miles, the applicable equation is y = 40 + 0.18 m .
Substitute m = 150 into the equation: y = 40 + 0.18 ( 150 ) .
Calculate the result: y = 40 + 27 = 67 .
The total amount owed is $67.00 .
Explanation
Understanding the Problem The problem states that a rental car company charges a base fee of $40 plus $0.25 per mile for the first 100 miles and $0.18 per mile for cars driven over 100 miles. We are given the piecewise function:
100\end{array}\right."> y = { 40 + 0.25 m , m ≤ 100 40 + 0.18 m , m > 100
We need to find the total amount owed if someone drives 150 miles.
Choosing the Correct Equation Since the person drives 150 miles, which is greater than 100 miles, we will use the second part of the piecewise function:
y = 40 + 0.18 m
Substituting the Value of m Now, we substitute m = 150 into the equation:
y = 40 + 0.18 ( 150 )
Calculating the Total Amount Calculating the value:
y = 40 + 0.18 × 150 = 40 + 27 = 67
So, the total amount owed is $67.00 .
Examples
Piecewise functions are useful in real life for modeling situations where different rules or conditions apply over different intervals. For example, cell phone plans often have a fixed monthly fee for a certain amount of data, and then charge a different rate for additional data used beyond that limit. Similarly, income tax brackets are structured using a piecewise function, where different tax rates apply to different income ranges. Understanding piecewise functions helps in making informed decisions about these types of plans and structures.
