24. you are organizing a school fair and rent a popcorn machine for 3 days. the rental company charges $65…

24. you are organizing a school fair and rent a popcorn machine for 3 days. the rental company charges $65 for the first day and $35 for each additional day. write and graph a step function that represents the relationship between the number x of days and the total cost y (in dollars) of renting the popcorn machine.

24. you are organizing a school fair and rent a popcorn machine for 3 days. the rental company charges $65 for the first day and $35 for each additional day. write and graph a step function that represents the relationship between the number x of days and the total cost y (in dollars) of renting the popcorn machine.

Answer

Explanation:

Step1: Analyze the cost for different number of days

When (x = 1), the cost (y=65). When (x = 2), the cost (y=65 + 35=100). When (x = 3), the cost (y=65+35\times2 = 65 + 70=135).

In general, for (x) days ((x) is a positive - integer), the step - function can be written as: [y=\begin{cases}65, &1\leq x < 2\100, &2\leq x < 3\135, &3\leq x\end{cases}]

Step2: Graph the step - function

  • For the interval ([1,2)), we have a horizontal line segment at (y = 65) (including (x = 1) and not including (x = 2)).
  • For the interval ([2,3)), we have a horizontal line segment at (y = 100) (including (x = 2) and not including (x = 3)).
  • For (x\geq3), we have a horizontal line segment at (y = 135) (including (x = 3) and all (x>3)).

Answer:

The step - function is (y=\begin{cases}65, &1\leq x < 2\100, &2\leq x < 3\135, &3\leq x\end{cases})