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Chapter 3: Problem 29

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The problem Maximize \(\quad P=x y\) $$ \text { subject to } \begin{aligned} & 2 x+3 y \leq 12 \\ & 2 x+y \leq 8 \\ & x \geq 0, y \geq 0 \end{aligned} $$ is a linear programming problem.

Short Answer

Expert verified
The statement is false because the objective function, P = xy, is not linear. While the constraints are linear inequalities and there are non-negativity constraints on the variables, a linear programming problem requires a linear objective function.

Step by step solution

01

Objective Function Linearity

In the given problem, the objective function is P = xy. This is not a linear function, as it is a product of x and y.
02

Constraints Linearity

The constraints in the problem are: 1. \(2x + 3y \leq 12\) 2. \(2x + y \leq 8\) These constraints are both linear inequalities.
03

Non-negativity Constraints

The variables x and y both have a non-negativity constraint. Since the objective function is not linear, the given problem is not a linear programming problem. The statement is #tag_underline#false.

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Most popular questions from this chapter

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

A farmer plans to plant two crops, A and B. The cost of cultivating crop A is \(\$ 40\) /acre whereas that of crop \(\mathrm{B}\) is \(\$ 60\) /acre. The farmer has a maximum of \(\$ 7400\) available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop \(\underline{B}\) requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. If she expects to make a profit of \(\$ 150\) /acre on crop \(\mathrm{A}\) and \(\$ 200\) acre on crop \(\mathrm{B}\), how many acres of each crop should she plant in order to maximize her profit? What is the optimal profit?

Solve each linear programming problem by the method of corners. $$ \begin{aligned} \text { Minimize } & C=2 x+10 y \\ \text { subject to } & 5 x+2 y \geq 40 \\ & x+2 y \geq 20 \\ & y \geq 3, x \geq 0 \end{aligned} $$

Perth Mining Company operates two mines for the purpose of extracting gold and silver. The Saddle Mine costs \(\$ 14,000 /\) day to operate, and it yields 50 oz of gold and 3000 oz of silver per day. The Horseshoe Mine costs \(\$ 16,000 /\) day to operate, and it yields 75 oz of gold and 1000 ounces of silver per day. Company management has set a target of at least 650 oz of gold and 18,000 oz of silver a. How many days should each mine be operated so that the target can be met at a minimum cost? b. Find the range of values that the Saddle Mine's daily operating cost can assume without changing the optimal solution. c. Find the range of values that the requirement for gold can assume. d. Find the shadow price for the requirement for gold.

Solve each linear programming problem by the method of corners. $$ \begin{aligned} \text { Maximize } & P=2 x+5 y \\ \text { subject to } & 2 x+y \leq 16 \\ & 2 x+3 y \leq 24 \\ y & \leq 6 \\ & x \geq 0, y & \geq 0 \end{aligned} $$
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