Answer:
(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: