The Donaldson Furniture Company produces three types of rocking chairs the child

Question

The Donaldson Furniture Company produces three types of rocking chairs the children's model, the standard model and the executive modal. Each char is made in three stages cutting construction, and finishing. The time needed for each stage of each char is given in the chart. During a specific week the company has available a maximum of 131 hours for cutting, 81 hours for construction, and 64 hours for finishing. Determine how many of each type of chair the company should make to be operating at full capacity. The number of executive chairs the company should make is

Explanation / Answer

We will solve this problem by setting up a system of linear equations which can be solved using the method of substitution or elimination.

Let x- denotes the number of childrens model chairs,

y- denotes the number of standard model chairs and

z- denotes the number of executive model chairs.

So, let's get things set up. We want to find the values of x, y, z, subject to given constraints. Chairs are constructed in three stages. In a particular week maximum hours available for cutting stage is 131 hours, for construction 81 hours and for finishing 64 hours.

Assume that company operates at its full capacity.

The total hours needed for cutting stage is 131 hours: The total hours needed for cutting the x children's model chairs would be 5x. The total hours needed for cutting the y standard model chairs would be 4y. The total hours needed for cutting the z executive model chairs would be 7z.

Thus, we get 5x+4y+7z=131 ...................(1).

Similarly, for construction stage we get the following equation:

3x+2y+5z=81 .............................(2)

And, for finishing stage we get the equation:

2x+2y+4z=64 ...........................(3).

We solve the above three equations simultaneuously using the method of elimination.

System of linear equations is

5x+4y+7z=131

3x+2y+5z=81

2x+2y+4z=64.

Multiply the second equation by 2 to get 6x+4y+10z=162 and subtract it from equation (1) we get:

5x+4y+7z-(6x+4y+10z)=131-162.

Simplifying it we get,

-x-3z=-31 which is same as x+3z=31.................(4).

Subtract equation (3) from equation (2) we get:

3x+2y+5z-(2x+2y+4z)=81-64. Simiplying this equation we get the following:

x+z=17 ................(5).

Now, we solve the equation (4) and (5) simultaneously.

Subtracting equation (5) from equation (4) we get:

x+3z-(x+z)=31-17.

Simplifying it we get, 2z=14 and therefore z=7.

Substituting the value of z=7 in equation (5) we get x+7=17 and hence x=10.

Again substitute z=7 and x=10 in equation (3) we get 2*10+2y+4*7=64.

Thus, 20+2y+28=64 which is same as 48+2y=64.

Hence, 2y=64-48 whcih gives 2y=16.

Therfore, y=8.   

Thus, the number of executive chairs company should make is 7.

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