Answer :
To determine which equation will create the given graph for the revenue earned when [tex]\( x \)[/tex] rooms are rented, let's analyze the provided equations step-by-step.
### Part A: Identifying the Correct Equation
#### Equation \#1: [tex]\( y = x^2 + 3x + 90 \)[/tex]
This equation is a quadratic equation where the coefficient of [tex]\( x^2 \)[/tex] is positive. This indicates that the parabola opens upwards. An upwards-opening parabola suggests that as [tex]\( x \)[/tex] increases or decreases from the vertex, the revenue will increase at both ends, which does not make sense for a realistic revenue scenario, where we expect a maximum revenue at some point rather than it increasing indefinitely. Therefore, this equation is less likely to represent our revenue model.
#### Equation \#2: [tex]\( y = -3x^2 + 90x \)[/tex]
This is another quadratic equation, but here the coefficient of [tex]\( x^2 \)[/tex] is negative. This signifies that the parabola opens downwards, which aligns with the business scenario of finding a peak revenue at a certain number of rented rooms. To confirm, we can look for the vertex of this parabola, which represents the maximum revenue point:
- For a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex], the x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- In this equation, [tex]\( a = -3 \)[/tex] and [tex]\( b = 90 \)[/tex].
So, the x-coordinate of the vertex is:
[tex]\[ x = -\frac{90}{2 \cdot (-3)} = 15 \][/tex]
The y-coordinate of the vertex can be found by substituting [tex]\( x = 15 \)[/tex] into the equation:
[tex]\[ y = -3(15)^2 + 90(15) = 675 \][/tex]
This gives us the vertex [tex]\((15, 675)\)[/tex], suggesting that the maximum revenue is 675 and occurs when 15 rooms are rented. This matches the characteristics of a realistic revenue graph with a peak value.
#### Equation \#3: [tex]\( y = (x-3)(x-30) \)[/tex]
Rewriting this equation in standard quadratic form, we get:
[tex]\[ y = x^2 - 33x + 90 \][/tex]
This is a quadratic equation where the coefficient of [tex]\( x^2 \)[/tex] is positive, making it an upwards-opening parabola. Similar to Equation \#1, this suggests that the revenue would increase indefinitely at both ends, which is not realistic for our scenario. Hence, this equation is less likely to represent the revenue model.
Given the analysis, the most suitable equation that fits the described scenario is:
Equation \#2: [tex]\( y = -3x^2 + 90x \)[/tex].
### Part B: Parts of the Equation that Influenced the Decision
The key parts of the equation that helped in making the decision are:
1. The Sign of the Quadratic Term ([tex]\( x^2 \)[/tex]): In Equation \#2, the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-3\)[/tex]). This indicates that the parabola opens downwards, which is crucial for modeling a scenario where there is a peak (maximum revenue) as opposed to the revenue increasing indefinitely.
2. The Vertex Calculation: The vertex of the parabola ([tex]\((15, 675)\)[/tex]), found by using the formulas for the x and y coordinates of the vertex for a quadratic equation, confirmed that there is a realistic peak revenue point at [tex]\( x = 15 \)[/tex] rooms, with a revenue of 675. This fits a real-world revenue model, where the maximum revenue can be identified.
In conclusion, the correct equation is Equation \#2: [tex]\( y = -3x^2 + 90x \)[/tex].
### Part A: Identifying the Correct Equation
#### Equation \#1: [tex]\( y = x^2 + 3x + 90 \)[/tex]
This equation is a quadratic equation where the coefficient of [tex]\( x^2 \)[/tex] is positive. This indicates that the parabola opens upwards. An upwards-opening parabola suggests that as [tex]\( x \)[/tex] increases or decreases from the vertex, the revenue will increase at both ends, which does not make sense for a realistic revenue scenario, where we expect a maximum revenue at some point rather than it increasing indefinitely. Therefore, this equation is less likely to represent our revenue model.
#### Equation \#2: [tex]\( y = -3x^2 + 90x \)[/tex]
This is another quadratic equation, but here the coefficient of [tex]\( x^2 \)[/tex] is negative. This signifies that the parabola opens downwards, which aligns with the business scenario of finding a peak revenue at a certain number of rented rooms. To confirm, we can look for the vertex of this parabola, which represents the maximum revenue point:
- For a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex], the x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- In this equation, [tex]\( a = -3 \)[/tex] and [tex]\( b = 90 \)[/tex].
So, the x-coordinate of the vertex is:
[tex]\[ x = -\frac{90}{2 \cdot (-3)} = 15 \][/tex]
The y-coordinate of the vertex can be found by substituting [tex]\( x = 15 \)[/tex] into the equation:
[tex]\[ y = -3(15)^2 + 90(15) = 675 \][/tex]
This gives us the vertex [tex]\((15, 675)\)[/tex], suggesting that the maximum revenue is 675 and occurs when 15 rooms are rented. This matches the characteristics of a realistic revenue graph with a peak value.
#### Equation \#3: [tex]\( y = (x-3)(x-30) \)[/tex]
Rewriting this equation in standard quadratic form, we get:
[tex]\[ y = x^2 - 33x + 90 \][/tex]
This is a quadratic equation where the coefficient of [tex]\( x^2 \)[/tex] is positive, making it an upwards-opening parabola. Similar to Equation \#1, this suggests that the revenue would increase indefinitely at both ends, which is not realistic for our scenario. Hence, this equation is less likely to represent the revenue model.
Given the analysis, the most suitable equation that fits the described scenario is:
Equation \#2: [tex]\( y = -3x^2 + 90x \)[/tex].
### Part B: Parts of the Equation that Influenced the Decision
The key parts of the equation that helped in making the decision are:
1. The Sign of the Quadratic Term ([tex]\( x^2 \)[/tex]): In Equation \#2, the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-3\)[/tex]). This indicates that the parabola opens downwards, which is crucial for modeling a scenario where there is a peak (maximum revenue) as opposed to the revenue increasing indefinitely.
2. The Vertex Calculation: The vertex of the parabola ([tex]\((15, 675)\)[/tex]), found by using the formulas for the x and y coordinates of the vertex for a quadratic equation, confirmed that there is a realistic peak revenue point at [tex]\( x = 15 \)[/tex] rooms, with a revenue of 675. This fits a real-world revenue model, where the maximum revenue can be identified.
In conclusion, the correct equation is Equation \#2: [tex]\( y = -3x^2 + 90x \)[/tex].
