The Donaldson Furniture Company produces three types of rocking chairs: the children's model, the standard model, and the executive model. Each chair is made in three stages: cutting, construction, and finishing. Stage Children's Standard Executive Cutting 5 hr 4 hr 7 hr Construction 3 hr 2 hr 5 hr Finishing 2 hr 2 hr 4 hr The time needed for each stage of each chair is given in the chart. During a specific week the company has available a maximum of 162 hours for cutting, 104 hours for construction, and 84 hours for finishing. Determine how many of each type of chair the company should make to be operating at full capacity.

Children's model = 7

Standard model = 9

Executive model = 13

Explanation:

Let a = number of children's model

b= number of standard model

c = number of executive model

According to the question given, the time allocation is as follows;

Cutting; a = 5 hr     b = 4hr     c = 7hr, Maximum hours = 162 hours

Construction; a = 3 hr     b = 2hr     c = 5hr, Maximum hours = 104 hours

Finishing; a = 2 hr     b = 2hr     c = 4hr, Maximum hours = 84 hours

We can therefore derive the equations.

5a + 4b + 7c = 162     (i)

3a + 2b + 5c = 104     (ii)

2a + 2b + 4c = 84     (iii)

Subtracting (iii) from (ii), a + c = 20, a = 20 - c     (iv)

Substituting (iv) into (ii)

60 - 3c + 2b + 5c = 104

2b + 2c = 44     (v)

Substituting (iv) into (i)

100 - 5c + 4b + 7c = 162

4b + 2c = 62     (vi)

Subtracting (v) from (vi), we have

2b = 18; b = 9

Since b + c = 22,

c =22 - 9 = 13

Also, a + c =20

a = 20 - 13 = 7