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={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+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={15+20x, 0<x≤312, x>3
y equals left brace line 1 15 plus 20 times x 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

Answer 1

To model the cost of renting a bouncy house as a piecewise function, we can outline the two scenarios based on the number of hours rented:

1. For rentals of **3 hours or less** (i.e., \( 0 < x \leq 3 \)):
- The cost is \( y = 15x + 20 \) (where \( x \) is the number of hours).

2. For rentals of **more than 3 hours** (i.e., \( x > 3 \)):
- The cost is \( y = 12x \).

Thus, we can represent this as a piecewise function:

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

Now, let's match this with the options given in your responses:

1. \( y = \{ 15x + 20, 0 < x \leq 3; 12x, x > 3 \} \)
2. \( y = \{ 15 + 20, 0 < x \leq 3; 12, x > 3 \} \)
3. \( y = \{ 15 + 20x, 0 < x \leq 3; 12, x > 3 \} \)
4. \( y = \{ 15x + 20, 0 < x < 3; 12x, x \geq 3 \} \)

The correct response is clearly the first one:

\[
y = \{ 15x + 20, 0 < x \leq 3; 12x, x > 3 \}
\]