Develop the Cost function.

The total cost function $C(x)$ is the sum of fixed costs and variable costs.
Fixed Cost (FC) = ₦10,000
Variable Cost per bottle (VC per unit) = ₦0.03
Let $x$ be the number of bottles produced.

The cost function is:

$$C(x) = \text{FC} + (\text{VC per unit}) \times x$$ $$C(x) = 10,000 + 0.03x$$

The capacity limitation is 50,000 bottles per month or 600,000 bottles per year.

Substitute $x = 600,000$ into the cost function. This quantity is within the annual capacity limit.

$$C(600,000) = 10,000 + 0.03 \times 600,000$$ $$C(600,000) = 10,000 + 18,000$$ $$C(600,000) = 28,000$$

The cost to produce 600,000 bottles is ₦28,000.

To plot $C(x) = 10,000 + 0.03x$:

• Draw a graph with the number of bottles ($x$) on the horizontal axis and Total Cost ($C(x)$) on the vertical axis.
• The y-intercept is the fixed cost: $(0, 10,000)$.
• Calculate a second point, for example, at the annual capacity limit: $(600,000, 28,000)$.
• Draw a straight line connecting these two points.

The revenue function is given as $R(x) = 0.46x$.
To plot $R(x) = 0.46x$ on the same graph:

• The line starts from the origin: $(0, 0)$.
• Calculate a second point, for example, at the annual capacity limit: $$R(600,000) = 0.46 \times 600,000 = 276,000$$ So, the point is $(600,000, 276,000)$.
• Draw a straight line connecting these two points.

The break-even point occurs when Total Revenue equals Total Cost ($R(x) = C(x)$).

$$0.46x = 10,000 + 0.03x$$

Subtract $0.03x$ from both sides:

$$0.46x - 0.03x = 10,000$$ $$0.43x = 10,000$$

Divide by 0.43:

$$x = \frac{10,000}{0.43}$$ $$x \approx 23,255.81$$

Since the number of bottles must be a whole number, we round up to ensure all costs are covered.

$$x = 23,256 \text{ bottles}$$

Now, calculate the Total Cost and Total Revenue at this quantity:

$$C(23,256) = 10,000 + 0.03 \times 23,256$$ $$C(23,256) = 10,000 + 697.68$$ $$C(23,256) = 10,697.68$$ $$R(23,256) = 0.46 \times 23,256$$ $$R(23,256) = 10,697.76$$

The slight difference in cost and revenue is due to rounding the number of bottles. Using the exact fraction for $x$ would yield identical values.

The break-even point is:

• Number of bottles: $\boxed{\text{23,256 bottles}}$
• Total Cost at break-even: $\boxed{\text{₦10,697.68}}$
• Total Revenue at break-even: $\boxed{\text{₦10,697.76}}$