Let's delve into part (d) in more detail.

### Problem Recap:
You need to determine the minimum number of bars of soap that the factory must produce in a day to avoid making a loss, i.e., to break even or make a profit.

### Understanding the Break-Even Point:
The break-even point occurs when the total revenue equals the total cost. At this point, the factory makes neither a profit nor a loss.

### Given:
- Selling price per bar of soap = £2
- Cost equation: \( y = 0.84x + 252 \), where:
  - \( 0.84x \) is the variable cost (cost per bar of soap).
  - £252 is the fixed cost (the cost that doesn't change with the number of bars made).

### Revenue Function:
The total revenue \( R \) from selling \( x \) bars of soap is:
\[
R = 2x
\]
where \( x \) is the number of bars produced and sold.

### Break-Even Condition:
For the factory to break even:
\[
\text{Revenue} = \text{Cost}
\]
\[
2x = 0.84x + 252
\]

### Solving for \( x \):
Subtract \( 0.84x \) from both sides:
\[
2x - 0.84x = 252
\]
\[
1.16x = 252
\]
Divide both sides by 1.16 to solve for \( x \):
\[
x = \frac{252}{1.16} \approx 217.24
\]

### Interpretation:
- The calculation shows that the factory needs to produce approximately 217.24 bars of soap to cover its costs exactly. 
- Since the number of bars of soap must be a whole number (you can't produce a fraction of a bar), the factory must produce at least **218 bars of soap** to ensure it doesn't make a loss and begins to make a profit.

### Conclusion:
To summarize, the least number of bars of soap that must be produced on any given day to avoid a loss is **218**. If the factory produces and sells at least 218 bars of soap, its revenue will cover both the fixed costs and the variable costs.

Would you like further clarification or have additional questions?

Here are 5 related questions to consider:

1. How would the minimum number of bars required change if the fixed cost increased to £300?
2. What happens to the break-even point if the selling price drops to £1.80 per bar?
3. How would the break-even point change if the variable cost per bar increased to £1.00?
4. If the factory can sell more bars, how would you calculate the profit for producing 500 bars?
5. How does the break-even analysis change if some bars are sold at a discounted rate?

**Tip:** Remember, the break-even point is where total revenue equals total costs. Any production beyond this point leads to profit, and any production below this point leads to a loss.

Explore the break-even analysis for a soap factory's production. Learn how to calculate the minimum production needed to avoid losses and achieve profitability. Detailed step-by-step solution provided.

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