Question

A model is used to predict monthly revenue for a new sports drink. In the model, x is the number of $1-price increases from the original $2 per bottle price. Model A uses points (9,605),(8,600),(10,600),(7,585) and (11,585).

Using Model A, the price should be $___ to maximize monthly revenue because vertex/y-intercept/x-intercept occurs at x=___

Answer 1

To determine the price that maximizes monthly revenue, we first need to find the vertex of the model represented by the points given. Let's assume these points represent a quadratic function \( R(x) = ax^2 + bx + c \), where \( R(x) \) is the revenue and \( x \) is the number of $1-price increases from the original $2 price.

### 1. Organize the points:
- The price of the sports drink becomes \( 2 + x \).
- The corresponding revenues from the points are:
- For \( x = 9 \), revenue = 605
- For \( x = 8 \), revenue = 600
- For \( x = 10 \), revenue = 600
- For \( x = 7 \), revenue = 585
- For \( x = 11 \), revenue = 585

### 2. Identify the maximum revenue:

From the revenues listed:
- Revenue at \( x = 7 \): 585
- Revenue at \( x = 8 \): 600
- Revenue at \( x = 9 \): 605 (maximum)
- Revenue at \( x = 10 \): 600
- Revenue at \( x = 11 \): 585

The maximum revenue occurs at \( x = 9 \).

### 3. Find the price that corresponds to this \( x \):
The price per bottle when \( x = 9 \) is:
\[
\text{Price} = 2 + 9 = 11 \text{ dollars}
\]

### 4. Determine the vertex:
In a quadratic function \( R(x) = ax^2 + bx + c \), the vertex \( x \) coordinate is found using the formula:
\[
x = -\frac{b}{2a}
\]
Since we only have points, we could graph the points to find the vertex visually, or as shown above, we’ve already found that the maximum revenue occurs at \( x = 9 \).

### Conclusion:
The price should be **$11** to maximize monthly revenue, and the vertex occurs at \( x = 9 \).

Therefore, the final answer is:
- The price should be **$11** to maximize monthly revenue because the vertex occurs at \( x = 9 \).