Question 1174395
Let's break down this problem step-by-step.

**(a) Breakeven Point**

1.  **Cost Function:**
    * Variable Cost (VC) = 10x
    * Fixed Cost (FC) = 6200
    * Total Cost (C(x)) = VC + FC = 10x + 6200

2.  **Revenue Function:**
    * R(x) = -2x² + 260x + 500

3.  **Breakeven Point:**
    * Breakeven occurs when Revenue (R(x)) = Cost (C(x)).
    * -2x² + 260x + 500 = 10x + 6200
    * -2x² + 250x - 5700 = 0
    * Divide by -2: x² - 125x + 2850 = 0

4.  **Solve the Quadratic Equation:**
    * We can use the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
    * In this case, a = 1, b = -125, c = 2850.
    * x = [125 ± √((-125)² - 4 * 1 * 2850)] / 2
    * x = [125 ± √(15625 - 11400)] / 2
    * x = [125 ± √4225] / 2
    * x = [125 ± 65] / 2

    * x1 = (125 + 65) / 2 = 190 / 2 = 95
    * x2 = (125 - 65) / 2 = 60 / 2 = 30

    * **Answer (a):** The breakeven points are at production levels of 30 units and 95 units.

**(b) Range of Production for Profit**

1.  **Profit Function:**
    * Profit (P(x)) = R(x) - C(x)
    * P(x) = (-2x² + 260x + 500) - (10x + 6200)
    * P(x) = -2x² + 250x - 5700

2.  **Profit Occurs When P(x) > 0:**
    * -2x² + 250x - 5700 > 0
    * x² - 125x + 2850 < 0 (divide by -2 and reverse the inequality)

3.  **Use the Breakeven Points:**
    * We already found the breakeven points at x = 30 and x = 95.
    * Since the parabola opens downward (-2x²), the profit is positive between the breakeven points.

    * **Answer (b):** The factory should produce between 30 and 95 units to have a profit (30 < x < 95).

**(c) Range of Production for Loss > RM 500**

1.  **Loss Function:**
    * Loss occurs when P(x) < 0.
    * We want to find when P(x) < -500.

2.  **Set Up Inequality:**
    * -2x² + 250x - 5700 < -500
    * -2x² + 250x - 5200 < 0
    * x² - 125x + 2600 > 0

3.  **Solve the Quadratic Equation:**
    * x² - 125x + 2600 = 0
    * Use the quadratic formula:
        * x = [125 ± √((-125)² - 4 * 1 * 2600)] / 2
        * x = [125 ± √(15625 - 10400)] / 2
        * x = [125 ± √5225] / 2
        * x = [125 ± 72.28] / 2

    * x1 = (125 + 72.28) / 2 ≈ 98.64
    * x2 = (125 - 72.28) / 2 ≈ 26.36

4.  **Determine the Range:**
    * Since the parabola opens upward, the inequality is satisfied when x < 26.36 or x > 98.64.

    * **Answer (c):** The factory has a loss greater than RM 500 when the number of productions is less than 26.36 units or greater than 98.64 units (x < 26.36 or x > 98.64).