A company charges $15 per hour plus a $20 setup fee to rent a bouncy house for 3 hours or less. They charge $12 per hour and no setup fee if the bouncy house is rented for more than 3 hours. Sketch a graph of the piecewise function that models this scenario. Use the graph to select the correct equation for the function where x

Question

is the number of hours and y
is the total cost.(1 point)
Responses

y={15+20, 0<x≤312, x>3
y equals left brace line 1 15 plus 20 comma 0 less than x less than or equal to three line 2 12 comma x greater than 3

y={15x+20, 0<x<312x, x≥3
y equals left brace line 1 15 times x + 20 comma 0 less than x less than three line 2 12 times x comma x greater than or equal to 3

y={15x+20, 0<x≤312x, x>3
y equals left brace line 1 15 times x plus 20 comma 0 less than x less than or equal to three line 2 12 times x comma x greater than 3

y={15+20x, 0<x≤312, x>3

Answer 1

To model the scenario described with a piecewise function, we need to define the cost \( y \) based on the number of hours \( x \):

1. If \( 0 < x \leq 3 \): The cost is calculated as \( 15x + 20 \). This includes the hourly rate of $15 times the number of hours plus a $20 setup fee.
2. If \( x > 3 \): The cost is calculated as \( 12x \). This reflects the hourly rate of $12 with no setup fee.

Based on the information provided, the piecewise function can be defined as follows:

\[ y = \begin{cases} 15x + 20 & 0 < x \leq 3 \ 12x & x > 3 \end{cases} \]

Now, let's look at the options provided:

1. y = {15 + 20, 0<x≤3; 12, x>3}

   • This is incorrect because it does not include \( x \) in the first part.

2. y = {15x + 20, 0<x<3; 12x, x≥3}

   • This is incorrect because it does not include \( x = 3 \) in the first part.

3. y = {15x + 20, 0<x≤3; 12x, x>3}

   • This is correct. It accurately captures the piecewise function, including \( x = 3 \) in the first part.

4. y = {15 + 20x, 0<x≤3; 12, x>3}

   • This is incorrect because it misrepresents the equations.

Thus, the correct equation for the piecewise function that models the scenario is:

y = {15x + 20, 0<x≤3; 12x, x>3}