A small factory makes bars of soap - Edexcel - A-Level Maths Pure - Question 9 - 2019 - Paper 2

Using the information provided:

1. For 800 bars sold, the revenue is: $800 imes 2 = £1600$ Substituting into the profit equation: $Profit = Revenue - Cost \Rightarrow 500 = 1600 - (k + 800c)$ This simplifies to: $k + 800c = 1100 \quad (1)$

2. For 300 bars sold, the revenue is: $300 imes 2 = £600$ Using the profit equation: $Loss = Cost - Revenue \Rightarrow 80 = (k + 300c) - 600$ This simplifies to: $k + 300c = 680 \quad (2)$

3. Now, we have two equations:

   • From (1): $k + 800c = 1100$
   • From (2): $k + 300c = 680$

4. Subtract equation (2) from equation (1): $(k + 800c) - (k + 300c) = 1100 - 680$ This gives: $500c = 420 \Rightarrow c = 0.84$

5. Substitute c into equation (2) to solve for k: $k + 300(0.84) = 680 \Rightarrow k + 252 = 680 \Rightarrow k = 428$

6. Therefore, we have: $y = 0.84x + 428$

The value 0.84 in the equation represents the variable cost of producing each additional bar of soap. This means that for each bar of soap produced, the factory incurs an additional cost of £0.84, which contributes to the overall production expenses.

To determine the break-even point, we need to set the profit to zero:

Profit = Revenue - Cost

Let's denote the number of bars sold as $n$. Therefore:

$2n - (k + cn) = 0 \Rightarrow 2n = k + cn$

Substituting the values of k and c:

$2n = 428 + 0.84n$

Rearranging gives:

$2n - 0.84n = 428 \Rightarrow 1.16n = 428 \Rightarrow n = \frac{428}{1.16} \approx 369.01$

Thus, rounding up, the least number of bars of soap that need to be made is:

370 bars.