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Chapter 3: Problem 15

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

Short Answer

Expert verified
The corner points of the feasible region are (1, 2), (0, 3), and (0, 2). Evaluating the objective function \(C = 3x + 4y\) at each corner point, we have \(C(1,2) = 11\), \(C(0,3) = 12\), and \(C(0,2) = 8\). The minimum value of C is 8, which occurs at the corner point (0, 2).

Step by step solution

01

Graph the constraints

Start by graphing each constraint on a coordinate plane: 1. \(x + y \geq 3\): Rewrite it as \(y \geq 3 - x\), and plot the line \(y = 3 - x\). Shade the region above the line to represent the inequality. 2. \(x + 2y \geq 4\): Rewrite it as \(y \geq \dfrac{1}{2} (4 - x)\), and plot the line \(y = \dfrac{1}{2}(4 - x)\). Shade the region above the line to represent the inequality. 3. \(x \geq 0\): This simply means that \(x\) is non-negative. Shade the region to the right of the y-axis. 4. \(y \geq 0\): This means that \(y\) is non-negative. Shade the region above the x-axis. The feasible region is the intersection of all shaded regions.
02

Find the corner points

Now we need to find the corner points of the feasible region. To do this, we will find the intersections of the constraint lines. There are three intersections to consider: 1. Intersection of \(x + y = 3\) and \(x + 2y = 4\): Solve the system of equations to find the intersection point. 2. Intersection of \(x + y = 3\) and \(x=0\): Plug \(x=0\) into the equation \(x+y=3\) and solve for \(y\). 3. Intersection of \(x + 2y = 4\) and \(x=0\): Plug \(x=0\) into the equation \(x+2y=4\) and solve for \(y\).
03

Evaluate the objective function at the corner points

Next, we need to evaluate the objective function \(C=3x+4y\) at each corner point to find which point gives the minimum value. 1. Corner Point 1: \(C(1,2)\) 2. Corner Point 2: \(C(0,3)\) 3. Corner Point 3: \(C(0,2)\)
04

Identify the minimum value

Finally, compare the C values obtained in Step 3 to determine which corner point gives the minimum value. The minimum value of C will be our answer.

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