D1 Linear Programming

Mr Baker is making cakes and fruit loaves for sale at a charity cake stall.
Each cake requires 200g of flour and 125g of fruit.
Each fruit loaf requires 200g of flour and 50g of fruit.
He has 2800g of flour and 1000g of fruit available.
Let the number of cakes that he makes be x and the number of fruit loaves he makes be y.

a) Show that these constraints can be modelled by the inequalities

x+y <= 14 and 5x+2y<= 40.

Each cake takes 50 minutes to cook and each fruit loaf takes 30 minutes to cook. There are 8 hours of cooking time available.

b) Obtain a further inequality, other than x>=0, y>=0, which models this time constraint.

c) On graph paper illustrate these three inequalities indicating clearly the feasible region.

It is decided to sell the cakes for £3.50 each and the fruit loaves for £1.50 each. Assuming that Mr Baker sells all that he makes,

d) Write down an expression for the amount of money P, in pounds, raised by the sale of Mr Baker's products.

e) Explaining your method clearly, determine how many cakes and how many fruit loaves Mr Baker should make in order to maximise P.

f) Write down the greatest value of P.