Step 1: 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. Step 2: Determine the cost to produce 600,000 bottles. 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. Step 3: Plot the relationship between Cost and the number of bottles produced. 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. Step 4: Plot the Revenue against output on the same graph. 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. Step 5: Determine the Cost, Revenue, and unit of bottles produced and sold at the break-even point. 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: $\boxed{\text{₦10,697.76}}$