A furniture manufacturing company manufactures dining room...

A furniture manufacturing company manufactures dining room tables and chairs. A table requires 8 labor-hours for assembling and 2 labor hours for finishing. A chair requires 2 labor-hours for assembling and 1 labor-hour for finishing. The maximum labor-hours available per day for assembly and finishing are 400 and $120,$ respectively. If $x$ is the number of tables and $y$ is the number of chairs produced per day, write a system of inequalities that indicates appropriate restrictions on $x$ and $y .$ Graph the set of feasible solutions for the number of tables and chairs that can be produced.

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Key Concepts

Graphical Representation

Graphical representation involves plotting the linear inequalities on a coordinate plane to visualize the feasible region. This method helps in understanding how the constraints interact and in identifying potential optimal solutions, especially in problems with two decision variables.

Linear Programming

Linear programming is a method used in optimization where the goal is to maximize or minimize a linear objective function subject to a set of linear constraints. This concept is widely used in resource allocation problems where limited resources, such as labor-hours, must be assigned in the most efficient way.

System of Linear Inequalities

A system of linear inequalities represents a collection of constraints that define the limits within which a solution must lie. In each inequality, a combination of decision variables is restricted to be less than or equal to (or greater than or equal to) a certain value, reflecting limitations such as maximum labor-hours available.

Feasible Region

The feasible region is the set of all points that satisfy the system of inequalities, representing all possible values of the decision variables that meet the constraints. This concept is crucial in linear programming as it identifies the domain over which the objective function is evaluated.

Transcript

00:01 Okay, for this problem, we have a furniture company, and we're dealing with dining room tables and chairs.

00:06 The dining room tables are going to be x, and the chairs is y.

00:12 And so we have to set up a system of inequalities that finds the number, basically the set of feasible solutions for how many tables and how many chairs are produced.

00:23 And this is broken up into labor hours for assembling and labor hours for finishing.

00:28 So we're going to start off with labor hours for assembling.

00:31 The tables require eight labor hours for assembling, so that would be eight times x because x is the tables.

00:39 And the chair requires two, so that's plus two y, for y being the chairs.

00:45 And the maximum hours for assembly is 400.

00:50 So maximum means you do less than or equal to 400.

00:53 You can go up to 400, but not more.

00:56 All right.

00:56 Now we're going to talk about the hours for finishing.

00:58 Okay, you have two hours for finishing.

01:01 A table in one hour for finishing a chair, so 2x plus 1y, and that total has to be less than or equal to 120.

01:11 So this represents our system.

01:14 Now we also are going to add in the fact that x has to be greater than or equal to zero, and y has to be greater than or equal to zero, because you cannot create negative chairs or negative tables...

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