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

We know the following:

1. When 800 bars are sold:

   • Selling price = 800 × £2 = £1600
   • Profit = £500
   • Total cost = £1600 - £500 = £1100

2. When 300 bars are sold:

   • Selling price = 300 × £2 = £600
   • Loss = £80
   • Total cost = £600 + £80 = £680

Using these equations:

1. For 800 bars: $1100 = k + 800c$ (Equation 1)

2. For 300 bars: $680 = k + 300c$ (Equation 2)

Now, we solve the two equations: From (1): $k + 800c = 1100$ From (2): $k + 300c = 680$

Subtract equation (2) from (1):

$800c - 300c = 1100 - 680$ $500c = 420$ $c = 0.84$

Now substitute c back into either equation to find k. Using (2):

$k + 300(0.84) = 680$ $k + 252 = 680$ $k = 680 - 252 = 428$

Thus, the equation becomes: $y = 0.84x + 428$.

In the context of the model, the value 0.84 represents the variable cost of making each additional bar of soap. This means that for every extra bar produced, the total cost of production increases by £0.84.

To find the least number of bars, we set the profit equation:

Profit = Revenue - Cost

We're looking for the point where:

Profit > 0

Revenue from selling x bars: $R = 2x$

Cost is: $C = k + cx = 428 + 0.84x$

Setting Profit to be greater than zero:

$2x - (428 + 0.84x) > 0$

Simplifying the inequality: $2x - 0.84x > 428$ $1.16x > 428$