Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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:

Step 2: 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).

Step 3: 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. \(C_A = 2(10) + 5(0) = 20\) \(C_B = 2(4) + 5(8) = 48\) \(C_C = 2(0) + 5(10) = 50\)

Step 4: 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 \(C = 20\).