Question

A cafe has just started to sell smoothies and wants to understand how it can change the price to make better total profit. The total profit (EP) in hundreds of pounds depends on the change in price (z) of each smoothie such that: P=12-4x-2x², where -5≤x≤5 (a) With reference to this model, give an interpretation of the 12 in the formula (b) Prove that there is no price change that will result in a total profit of £1600 You must show your reasoning clearly [1 ma [3 ma The cafe is said to 'break even' when the total profit is zero. Given that the current price of a smoothie £6 and the owner is looking to change this; (c) What prices would the cafe need to charge to be on the point of breaking even? ​

Answer 1

(a) The 12 in the formula represents the total profit in hundreds of pounds when the price change (x) is 0, i.e. when there is no change in price from the original £6.

(b) To prove there is no price change that results in a profit of £1600:

P = 12 - 4x - 2x^2

1600 = 12 - 4x - 2x^2

2x^2 + 4x - 1588 = 0

Using the quadratic formula:

x = (-4 ± √(4^2 - 42-1588)) / (2*2)

There are no real solutions for x.

Therefore, there is no real value of x (price change) that can make the profit 1600.

(c) To find the break even points, set P=0 and solve for x:

0 = 12 - 4x - 2x^2

2x^2 + 4x - 12 = 0

x = -6, 2

So the prices that cause the cafe to break even are £6 - £6 = £0 and £6 + £2 = £8.

Have a great day!

Explanation: