Chapter 11: Q. 52 (page 794)

In Problems 51 and 52, solve each linear programming problem.
Minimize $z = 3 x + 5 y$ when $x \geq 0; y \geq 0; 3 x + 2 y \leq 12; x + 3 y \leq 12$

Short Answer

Expert verified

By minimizing $z = 3 x + 5 y$ when $x \geq 0; y \geq 0; 3 x + 2 y \leq 12; x + 3 y \leq 12$ , the maximum value is $3$ which occurs at $x = 1; y = 0$

Step by step solution

Step 1. Given

The problem:

$z = 3 x + 5 y$

The constraints:

$x \geq 0; y \geq 0; 3 x + 2 y \leq 12; x + 3 y \leq 12$

Step 2. Graph the constraints.

The graph of the constraints (the feasible points) is illustrated:

Step 3. List the corner points

We list the corner points and evaluate the objective function at each.

S.No.	Corner points(x, y)	Value of the objective function $z = 3 x + 5 y$
1.	$(1, 0)$	$z = 3 (1) + 5 (0) = 3$
2.	$(4, 0)$	$z = 3 (4) + 5 (0) = 12$
3.	$(1.71, 3.43)$	$z = 3 (1.71) + 5 (3.43) = 22.1$
4.	$(0, 4)$	$z = 3 (0) + 5 (4) = 20$
5.	$(0, 1)$	$z = 3 (0) + 5 (1) = 5$

Step 4. Minimize the objective function.

From the table, the minimum value of the given objective function is $3$ and it occurs at $x = 1; y = 0$

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In Problems 51 and 52, solve each linear programming problem.
Minimize $z = 3 x + 5 y$ when $x \geq 0; y \geq 0; 3 x + 2 y \leq 12; x + 3 y \leq 12$

Short Answer

Step by step solution

Step 1. Given

Step 2. Graph the constraints.

Step 3. List the corner points

Step 4. Minimize the objective function.

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In Problems 51 and 52, solve each linear programming problem.Minimizez=3x+5y when x≥0;y≥0;3x+2y≤12;x+3y≤12

Short Answer

Step by step solution

Step 1. Given

Step 2. Graph the constraints.

Step 3. List the corner points

Step 4. Minimize the objective function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Statistics

Discrete Mathematics

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 51 and 52, solve each linear programming problem.
Minimize $z = 3 x + 5 y$ when $x \geq 0; y \geq 0; 3 x + 2 y \leq 12; x + 3 y \leq 12$