Here are the solutions to the questions: 2.2.1 Determine the total cost function $TC(Q)$. The total cost consists of a fixed cost and a variable cost. Fixed cost (FC) = R30. Variable cost per unit = R3. For $Q$ units, the total variable cost (VC) is $3Q$. The total cost function $TC(Q)$ is the sum of fixed cost and total variable cost. $$TC(Q) = FC + VC(Q)$$ $$TC(Q) = 30 + 3Q$$ The total cost function is $\boxed{TC(Q) = 3Q + 30}$. 2.2.2 Explain the value of $c$ in the profit function. The profit function is given as $\pi(Q) = aQ^2 + bQ + c$. Profit is defined as Total Revenue minus Total Cost: $\pi(Q) = TR(Q) - TC(Q)$. When the quantity produced $Q = 0$, there is no revenue, so $TR(0) = 0$. However, the fixed cost of R30 is still incurred, so $TC(0) = 30$. Therefore, the profit at $Q=0$ is $\pi(0) = TR(0) - TC(0) = 0 - 30 = -30$. From the profit function, $\pi(0) = a(0)^2 + b(0) + c = c$. Thus, $c = -30$. The value of $c$ represents the fixed cost incurred as a loss when no units are produced ($Q=0$). 2.2.3 Given that $\pi(2) = -8$ and $\pi(8) = 16$, determine $a$ and $b$. From 2.2.2, we know $c = -30$. So the profit function is $\pi(Q) = aQ^2 + bQ - 30$. Using the given information: 1. For $\pi(2) = -8$: $$a(2)^2 + b(2) - 30 = -8$$ $$4a + 2b - 30 = -8$$ $$4a + 2b = 22$$ Dividing by 2, we get: $$2a + b = 11 \quad (Equation \ 1)$$ 2. For $\pi(8) = 16$: $$a(8)^2 + b(8) - 30 = 16$$ $$64a + 8b - 30 = 16$$ $$64a + 8b = 46 \quad (Equation \ 2)$$ From Equation 1, express $b$ in terms of $a$: $$b = 11 - 2a$$ Substitute this into Equation 2: $$64a + 8(11 - 2a) = 46$$ $$64a + 88 - 16a = 46$$ $$48a = 46 - 88$$ $$48a = -42$$ $$a = \frac{-42}{48}$$ $$a = -\frac{7}{8}$$ Now substitute the value of $a$ back into the expression for $b$: $$b = 11 - 2\left(-\frac{7}{8}\right)$$ $$b = 11 + \frac{14}{8}$$ $$b = 11 + \frac{7}{4}$$ $$b = \frac{44}{4} + \frac{7}{4}$$ $$b = \frac{51}{4}$$ The values are $\boxed{a = -\frac{7}{8} \text{ and } b = \frac{51}{4}}$. 2.2.4 Determine the break-even point(s). Break-even points occur when profit $\pi(Q) = 0$. Using the profit function with $a = -\frac{7}{8}$, $b = \frac{51}{4}$, and $c = -30$: $$\pi(Q) = -\frac{7}{8}Q^2 + \frac{51}{4}Q - 30$$ Set $\pi(Q) = 0$: $$-\frac{7}{8}Q^2 + \frac{51}{4}Q - 30 = 0$$ Multiply the entire equation by 8 to eliminate fractions: $$-7Q^2 + 102Q - 240 = 0$$ Multiply by -1 to make the leading coefficient positive: $$7Q^2 - 102Q + 240 = 0$$ Use the quadratic formula $Q = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$: Here, $A=7$, $B=-102$, $C=240$. $$Q = \frac{-(-102) \pm \sqrt{(-102)^2 - 4(7)(240)}}{2(7)}$$ $$Q = \frac{102 \pm \sqrt{10404