Question
A) 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)
y =
15x + 20, 0 < x ≤ 3
{ 12x, x > 3
y =
15 + 20, 0 < x ≤ 3
{ 12, x > 3
y =
15x + 20, 0 < x < 3
{ 12x, x ≥ 3
y =
15 + 20x, 0 < x ≤ 3
{ 12, x > 3
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)
y =
15x + 20, 0 < x ≤ 3
{ 12x, x > 3
y =
15 + 20, 0 < x ≤ 3
{ 12, x > 3
y =
15x + 20, 0 < x < 3
{ 12x, x ≥ 3
y =
15 + 20x, 0 < x ≤ 3
{ 12, x > 3
Answers
GPT-4o mini
To model the scenario you provided, we need to define a piecewise function that represents the cost of renting the bouncy house based on the number of hours.
1. For rentals of 3 hours or less (0 < x ≤ 3), the cost is $15 per hour plus a $20 setup fee. The total cost y in this case can be represented as:
\[
y = 15x + 20
\]
2. For rentals of more than 3 hours (x > 3), the cost is $12 per hour with no setup fee. The total cost in this case is:
\[
y = 12x
\]
So, the piecewise function can be represented as:
\[
y =
\begin{cases}
15x + 20 & \text{for } 0 < x \leq 3 \\
12x & \text{for } x > 3
\end{cases}
\]
Now, let's analyze the options provided:
1. **Option 1:** \( y = 15x + 20, \, 0 < x \leq 3 \) \{ \( 12x \), \( x > 3 \) \}
This option matches the piecewise function we derived.
2. **Option 2:** \( y = 15 + 20, \, 0 < x \leq 3 \) \{ \( 12 \), \( x > 3 \) \}
This option is incorrect; it does not correctly relate to the cost structure.
3. **Option 3:** \( y = 15x + 20, \, 0 < x < 3 \) \{ \( 12x \), \( x \geq 3 \) \}
This option is incorrect because the first part should include \( x = 3 \) (not just 0 < x < 3).
4. **Option 4:** \( y = 15 + 20x, \, 0 < x \leq 3 \) \{ \( 12 \), \( x > 3 \) \}
This option is incorrect; it misrepresents the pricing structure.
Given this analysis, **the correct equation for the function that models this scenario is Option 1**:
\[
y =
\begin{cases}
15x + 20 & \text{for } 0 < x \leq 3 \\
12x & \text{for } x > 3
\end{cases}
\]
1. For rentals of 3 hours or less (0 < x ≤ 3), the cost is $15 per hour plus a $20 setup fee. The total cost y in this case can be represented as:
\[
y = 15x + 20
\]
2. For rentals of more than 3 hours (x > 3), the cost is $12 per hour with no setup fee. The total cost in this case is:
\[
y = 12x
\]
So, the piecewise function can be represented as:
\[
y =
\begin{cases}
15x + 20 & \text{for } 0 < x \leq 3 \\
12x & \text{for } x > 3
\end{cases}
\]
Now, let's analyze the options provided:
1. **Option 1:** \( y = 15x + 20, \, 0 < x \leq 3 \) \{ \( 12x \), \( x > 3 \) \}
This option matches the piecewise function we derived.
2. **Option 2:** \( y = 15 + 20, \, 0 < x \leq 3 \) \{ \( 12 \), \( x > 3 \) \}
This option is incorrect; it does not correctly relate to the cost structure.
3. **Option 3:** \( y = 15x + 20, \, 0 < x < 3 \) \{ \( 12x \), \( x \geq 3 \) \}
This option is incorrect because the first part should include \( x = 3 \) (not just 0 < x < 3).
4. **Option 4:** \( y = 15 + 20x, \, 0 < x \leq 3 \) \{ \( 12 \), \( x > 3 \) \}
This option is incorrect; it misrepresents the pricing structure.
Given this analysis, **the correct equation for the function that models this scenario is Option 1**:
\[
y =
\begin{cases}
15x + 20 & \text{for } 0 < x \leq 3 \\
12x & \text{for } x > 3
\end{cases}
\]