Question

In: Advanced Math

A company produces wooden chairs and tables. Each chair uses 10 lbs. of wood and takes...

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.

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

Expert Solution

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

Colby Messinger answered 2 years ago

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