Mike's Toy Truck Company manufactures two models of toy...

Mike's Toy Truck Company manufactures two models of toy trucks, a standard model and a deluxe model. Each standard model requires 2 hours (h) for painting and 3 h for detail work; each deluxe model requires $3 \mathrm{~h}$ for painting and $4 \mathrm{~h}$ for detail work. Two painters and three detail workers are employed by the company, and each works 40 h per week. (a) Using $x$ to denote the number of standard-model trucks and $y$ to denote the number of deluxe-model trucks, write a system of linear inequalities that describes the possible numbers of each model of truck that can be manufactured in a week. (b) Graph the system and label the corner points.

Mike's Toy Truck Company manufactures two models of toy trucks, a standard model and a deluxe model. Each standard model requires 2 hours (h) for painting and 3 h for detail work; each deluxe model requires $3 \mathrm{~h}$ for painting and $4 \mathrm{~h}$ for detail work. Two painters and three detail workers are employed by the company, and each works 40 h per week.
(a) Using $x$ to denote the number of standard-model trucks and $y$ to denote the number of deluxe-model trucks, write a system of linear inequalities that describes the possible numbers of each model of truck that can be manufactured in a week.
(b) Graph the system and label the corner points.

Key Concepts

System of Linear Inequalities

A system of linear inequalities involves multiple linear inequalities that together impose several conditions or limits on the variables involved. In production problems, each inequality represents a constraint such as the limited availability of resources like labor or material, ensuring that only feasible production combinations are considered.

Feasible Region

The feasible region is the set of all points that satisfy every inequality in the system simultaneously. It represents all possible solutions—such as different production combinations—that meet the imposed resource or capacity constraints, and is typically visualized on a graph.

Graphical Solution Method

Graphing a system of linear inequalities involves drawing the boundary lines for each inequality on a coordinate plane and then shading the regions that satisfy each inequality. The intersection of these shaded regions forms the feasible region, which helps in visualizing all viable solution options.

Corner Points (Vertices)

Corner points, or vertices, of the feasible region are the intersections of the boundary lines defined by the inequalities. These points are crucial because in linear programming problems, the optimal solution (e.g., maximum profit or minimum cost) is often found at one of these vertices.

Transcript

00:01 Okay, so we are going to start with the linear programming problem.

00:05 Millie's toy truck company manufactures two models of toy trucks, a standard model and a deluxe model.

00:10 Each standard model requires two hours of painting for three hours of detail work, and each deluxe model requires three hours of painting and four hours of detail work.

00:21 Two painters and three detail workers are employed by the company, and each works 40 hours per week.

00:28 Using x to denote the number of standard model trucks and y to denote the number of deluxe model trucks so let's write that down so x is the number of standard model and y is the number of deluxe model we are asked to write a system of linear inequalities that describes the possible numbers of each model that can be manufactured in a week and graph the system to label the corner points.

01:12 So we're first going to need our equation for painting.

01:21 So our painting equation is that our standard model requires two hours of painting.

01:32 So that is two hours for each standard model, so 2x plus the three hours for painting for the deluxe model.

01:46 And our painters, there are two painters who can each work 40 hours per week, which means that this has to be less than or equal to the 80 hours, those two have combined.

02:05 The second equation is the detail work, which the standard model requires three hours of detail work, so 3x, plus the four hours of detail work for the deluxe model, and that has to be less than equal to the 40 hours for the three detail workers, that is 120 hours.

02:41 So we also know that we have to make more than zero of each type, because we cannot make a negative number of trucks.

02:53 So we have exactly zero, but we cannot have less than zero.

02:57 So we need to graph this.

02:59 We are only going to be graphing this on the first quadrant because our x and y cannot be negative.

03:11 So if we solve for our intercepts or we could solve for slope intercept form, i personally prefer slope intercept form.

03:19 So if i solve this into slope intercept form, y is less than or equal to negative two thirds x plus 80 divided by three, which is 26 .6 .6.

03:53 We could also determine that our slope is negative two -thirds and our x intercept is 40...

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