2.2.1 Determine the total cost function TC(Q).

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, $A=7$, $B=-102$, $C=240$.

$$Q = \frac{-(-102) \pm \sqrt{(-102)^2 - 4(7)(240)}}{2(7)}$$ $$$$