You operate a small wooden toy company making two products – alphabet blocks and wooden trucks.
Your profit is $30 per box of blocks and $40 per box of trucks.
The production a box of blocks requires 1 hour of woodworking and 2 hours of painting.
The production of a box of trucks requires 3 hours of woodworking and 1 hour of painting.
You employ 3 woodworkers and 2 painters – each working 40 hours per week.
Use linear programming to determine how many boxes of blocks and how many boxes of trucks you would make each week to maximise profit.

the profit earned per week is $800

Explanation:

firstly we separate information given let X be the number of blocks and Y be the number of trucks.

For box of Blocks :

profit is $30 per box of blocks

we are told that 1 hour of woodworking is required and 2 hours of painting for production.

For box of Trucks:

profit is $40 per box of blocks

we are further told that 3 hours of woodworking are required and also 1 hour of painting for production.

then we are told 3 woodworkers and 2 painters are hired each working 40 hours per week.

then now we are creating equations:

the first equation is going to be the profit equation which is

Profit = $30X +$40Y  (i)

the second and third equation will be the time equations:

1X + 3Y = 40 (ii)  which means if an hour of a box of blocks worked and 3 hours of block trucks in woodworking must optimize to 40 hours per week.

2X + 1Y = 40 (iii)  which means if an 2 hours of a box of blocks worked and  an hour of block trucks worked in painting must optimize to 40 hours per week.

therefore we firstly get how many boxes are produced per week for Trucks(Y) and blocks (X) so from the above equations we make X or Y the subject of the formula:

from equation (ii)

X+3Y= 40 then we transpose 3Y,

X= 40 - 3Y (iv)  then we substitute this into equation (iii)

2(40 - 3Y) + Y = 40 then we solve for Y

80 - 6Y + Y = 40

-5Y= -40 divide both sides by -5

Y= 8 then we substitute this Y value to equation (iv)

X= 40 -3(8)

X = 16

so we now know that per week they produce 16 boxes of blocks and 8 boxes of trucks now we find the profit per week which we substitute these values to equation (i):

Profit per week = $30(16) + $40(8)

Profit per week = $800