Question

A company produces wooden chairs and tables. Each chair uses 10 lbs. of wood and takes 10 hours to construct. Each table uses 15 lbs. of wood and takes 5 hours to construct. There are up to 150 lbs. of wood available and up to 110 labor hours available. If the profit on each chair is $3 and the profit on each table is 4$. We wish to maximize the profit. Let x be the number of chairs produced and y be the number of tables produced.

1. What is the formula to be maximized?
2. Write down the set of linear inequalities associated with the situation. Note: You should recognize that x and y can’t be negative.
3. Graph the feasible region by hand.
4. Find the coordinates of each vertex of the feasible region
5. Use these vertices to find the associated maximum

Answer 1

Let   be the number of chairs produced  

Let   be the number of tables produced

The profit on each chair =  

The profit on each table =  

Therefore the total profit on   chairs and   tables =  

Now,

Wood required for each chair =  10 lbs

Wood required for each table =   15 lbs  

Maximum availability of wood = 150 lbs

Therefore for   chairs and   tables , we have ,

Again,

labour time for each chair =  10 hours

labour time for each table =  5 hours

Maximum available labour hour =  110

Therefore for   chairs and   tables , we have ,

Thus we have the maximisation(to maximise the profit) problem as :

Maximise :  

subject to the linear inequalities :

  

  

   ,

Now we can graph the feasible region :  

  

The shaded region OABC is the feasible region

where , O , A , B , and C are the vertices

Now the coordinates of vertices are : O(0,0) , A(0,10) , C(11,0)  

To find the coordinates of B we can solve

Therefore coordinates of : B(9,4)

Thus the vertices are :

O(0,0) , A(0,10) ,  B(9,4) , C(11,0)

Our aim is to maximise  

The values of   at the vertices :

O(0,0) = 0

A(0,10) =  40

B(9,4) = 43

C(11,0) = 33

Therefore the maximum profit occurs at   B(9,4) = $ 43

Thus to maximise the profit the company needs to produce 9 chairs and 4 tables