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Chapter 6: Problem 21

Solve each linear programming problem by the method of corners. $$ \begin{array}{rr} \text { Minimize } & C=2 x+5 y \\ \text { subject to } & 4 x+y \geq 40 \\ & 2 x+y \geq 30 \\ & x+3 y \geq 30 \\ & x \geq 0, y \geq 0 \end{array} $$

Short Answer

Expert verified
The minimum value of the objective function \(C = 2x + 5y\) is achieved at point A(10, 0), with a value of \(C = 20\).

Step by step solution

01

Identify the feasible region

First, we need to find the feasible region, which is the set of all points that satisfy all the constraints. Let's rewrite the inequalities as equations and plot the corresponding lines on a graph. Equality forms of the given inequalities: \(4x + y = 40\) \(2x + y = 30\) \(x + 3y = 30\) Now we determine the coordinates of the intersecting points:
02

Locate the corner points

To find the corner points, we'll solve the equations to find the points where the lines intersect. To do this, we can use the method of substitution or elimination. Intersection points of equations: \(A: \) intersection of \(4x + y = 40\) and \(2x + y = 30\) \(B: \) intersection of \(4x + y = 40\) and \(x + 3y = 30\) \(C: \) intersection of \(2x + y = 30\) and \(x + 3y = 30\) Solving these pairs of equations: \(A: (10, 0)\) \(B: (4, 8)\) \(C: (0, 10)\) So the corner points of the feasible region are A(10,0), B(4,8), and C(0,10).
03

Evaluate the objective function

Now, we'll evaluate the objective function \(C = 2x + 5y\) at each of the corner points A, B, and C to find the minimum value. \(C_A = 2(10) + 5(0) = 20\) \(C_B = 2(4) + 5(8) = 48\) \(C_C = 2(0) + 5(10) = 50\)
04

Choose the corner point with the minimum value

According to the results of Step 3, the minimum value of the objective function is at point A(10,0) with a value of \(C_A = 20\). Therefore, the solution of the linear programming problem is when \(x = 10\) and \(y = 0\), with a minimum objective function value of \(C = 20\).

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Most popular questions from this chapter

A company manufactures products \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Each product is processed in three departments: I, II, and III. The total available labor-hours per week for departments I, II, and III are 900,1080 , and 840 , respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows: $$ \begin{array}{lccc} \hline & \text { Product A } & \text { Product B } & \text { Product C } \\ \hline \text { Dept. I } & 2 & 1 & 2 \\ \hline \text { Dept. II } & 3 & 1 & 2 \\ \hline \text { Dept. III } & 2 & 2 & 1 \\ \hline \text { Profit } & \$ 18 & \$ 12 & \$ 15 \\ \hline \end{array} $$ How many units of each product should the company produce in order to maximize its profit? What is the largest profit the company can realize? Are there any resources left over?

Solve each linear programming problem by the simplex method. $$ \begin{aligned} \text { Maximize } & P=2 x+6 y+6 z \\ \text { subject to } & 2 x+y+3 z \leq 10 \\ & 4 x+y+2 z \leq 56 \\ & 6 x+4 y+3 z \leq 126 \\ & 2 x+y+z \leq 32 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=4 x+6 y \\ \text { subject to } & 3 x+y \leq 24 \\ & 2 x+y \leq 18 \\ & x+3 y \leq 24 \\ & x \geq 0, y \geq 0 \end{array} $$

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{rrrrr|c} x & y & u & v & P & \text { Constant } \\ \hline 1 & 1 & 1 & 0 & 0 & 6 \\ 1 & 0 & -1 & 1 & 0 & 2 \\ \hline 3 & 0 & 5 & 0 & 1 & 30 \end{array} $$

Solve each linear programming problem by the method of corners. Find the maximum and minimum of \(P=4 x+3 y\) subject to $$ \begin{aligned} 3 x+5 y & \geq 20 \\ 3 x+y & \leq 16 \\ -2 x+y & \leq 1 \\ x \geq 0, y & \geq 0 \end{aligned} $$
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