Question

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

Answer 1

To model the given scenario using a piecewise function, we define the function \( y \) (total cost) based on the rental time \( x \) (in hours).

1. For \( 0 < x \leq 3 \) (i.e., for 3 hours or less):
- The cost is $15 per hour plus a $20 setup fee.
- The total cost function in this case is:
\[
y = 15x + 20
\]

2. For \( x > 3 \) (i.e., for more than 3 hours):
- The cost is $12 per hour with no setup fee.
- The total cost function in this case is:
\[
y = 12x
\]

So the complete piecewise function is:
\[
y =
\begin{cases}
15x + 20 & \text{if } 0 < x \leq 3 \\
12x & \text{if } x > 3
\end{cases}
\]

Now, choosing from the options provided, the correct equation for the function is:
\[
y =
\begin{cases}
15x + 20 & 0 < x \leq 3 \\
12x & x > 3
\end{cases}
\]

This matches with:
**y={15x+20, 0<x≤3; 12x, x>3}**.

Note:
The option you might see as "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" has the right structure but might be expressed unclearly, so ensure that you recognize the correct configuration.