Math Problem Statement

Your salami manufacturing plant can order up to 1,000 pounds of pork and 2,400 pounds of beef per day for use in manufacturing its two specialties: Count Dracula Salami and Frankenstein Sausage. Production of the Count Dracula variety requires 1 pound of pork and 3 pounds of beef for each salami, while the Frankenstein variety requires 2 pounds of pork and 2 pounds of beef for every sausage. In view of your heavy investment in advertising Count Dracula Salami, you have decided that at least one-third of the total production should be Count Dracula. On the other hand, because of the health-conscious consumer climate, your Frankenstein Sausage (sold as having less beef) is earning your company a profit of $5 per sausage, while sales of the Count Dracula variety are down and earning only $1 per salami. Given these restrictions, how many of each kind of sausage should you produce to maximize profits, and what is the maximum possible profit (in dollars)?

Solution

To solve the problem, we need to create a linear programming model based on the information provided.

Decision Variables

  1. Let xx represent the number of Count Dracula Salamis produced.
  2. Let yy represent the number of Frankenstein Sausages produced.

Objective Function

Maximize profit: Profit=5y+1x\text{Profit} = 5y + 1x

Constraints

  1. Pork availability: x+2y1000x + 2y \leq 1000 (Count Dracula uses 1 pound of pork, and Frankenstein uses 2 pounds per unit.)

  2. Beef availability: 3x+2y24003x + 2y \leq 2400 (Count Dracula uses 3 pounds of beef, and Frankenstein uses 2 pounds per unit.)

  3. Minimum Count Dracula production: At least one-third of the total production should be Count Dracula: x13(x+y)x \geq \frac{1}{3}(x + y) Simplify: 2xy2x \geq y

  4. Non-negativity: x0,y0x \geq 0, \, y \geq 0

Solving the System

I will calculate the optimal solution to maximize the profit under these constraints.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization
Inequalities

Formulas

Profit = 5y + x
Pork constraint: x + 2y ≤ 1000
Beef constraint: 3x + 2y ≤ 2400
Minimum production constraint: 2x ≥ y
Non-negativity: x ≥ 0, y ≥ 0

Theorems

Feasible Region Theorem
Optimality of Linear Objective Functions

Suitable Grade Level

Grades 11-12 or College Level