Question 1176522
Let's solve this problem step-by-step.

**A. Model the Profit Function**

1.  **Find the Demand Function (p(x))**

    * We have two points: (600, 1.00) and (560, 1.25)
    * Find the slope (m):
        * m = (1.25 - 1.00) / (560 - 600) = 0.25 / -40 = -0.00625
    * Use point-slope form (y - y1 = m(x - x1)):
        * p - 1.00 = -0.00625(x - 600)
        * p = -0.00625x + 3.75 + 1.00
        * p(x) = -0.00625x + 4.75

2.  **Find the Revenue Function (R(x))**

    * Revenue = price * quantity
    * R(x) = x * p(x)
    * R(x) = x(-0.00625x + 4.75)
    * R(x) = -0.00625x² + 4.75x

3.  **Find the Cost Function (C(x))**

    * Cost = startup cost + cost per bag * quantity
    * C(x) = 500 + 0.50x

4.  **Find the Profit Function (P(x))**

    * Profit = Revenue - Cost
    * P(x) = R(x) - C(x)
    * P(x) = (-0.00625x² + 4.75x) - (500 + 0.50x)
    * P(x) = -0.00625x² + 4.25x - 500

**B. Maximize Profit**

1.  **Find the Vertex of the Profit Function**

    * The profit function is a quadratic, so its maximum occurs at the vertex.
    * The x-coordinate of the vertex is given by x = -b / 2a, where a = -0.00625 and b = 4.25.
    * x = -4.25 / (2 * -0.00625)
    * x = -4.25 / -0.0125
    * x = 340

2.  **Find the Price per Bag**

    * p(x) = -0.00625x + 4.75
    * p(340) = -0.00625(340) + 4.75
    * p(340) = -2.125 + 4.75
    * p(340) = 2.625

3.  **Find the Maximum Profit**

    * P(x) = -0.00625x² + 4.25x - 500
    * P(340) = -0.00625(340)² + 4.25(340) - 500
    * P(340) = -0.00625(115600) + 1445 - 500
    * P(340) = -722.5 + 1445 - 500
    * P(340) = 222.5

**Answers for B:**

* Number of bags: 340
* Price per bag: $2.625 (or $2.63)

**C. Agreement with Intuition**

* **Initial Intuition:** You might have thought that raising the price would always decrease the number of bags sold and potentially lower the profit.
* **Actual Result:** The analysis shows that there's an optimal price point that maximizes profit. In this case, raising the price significantly above the initial $1.00 leads to a lower quantity sold but a higher profit.
* **Explanation:** This is because the higher price per bag more than compensates for the reduced quantity sold, up to a certain point. The profit function is a parabola, and the vertex represents the optimal balance between price and quantity.

**Why the "thought" might not be what the answer is:**

* **Linear Demand Assumption:** The linear demand function is a simplification. Real-world demand might not be perfectly linear.
* **Cost Structure:** The constant cost per bag and fixed startup cost simplify the model. Real costs might be more complex.
* **Consumer Behavior:** The model assumes rational consumer behavior. In reality, factors like brand loyalty, perceived value, and competitor pricing can influence demand.

In conclusion, the mathematical analysis provides a more precise and optimal solution than relying on initial intuition alone.