Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Optimization Problem for Food Production**
A biologist has 600 kg of nutrient A, 800 kg of nutrient B, and 500 kg of nutrient C. These nutrients will be used to make 4 types of food, whose contents (in percent per kilogram of food) and whose 'growth values' are shown in the table below.
| | P | Q | R | S |
|-----|-----|-----|-----|-----|
| A | 0 | 0 | 12.5 | 37.5 |
| B | 0 | 20 | 37.5 | 62.5 |
| C | 100 | 80 | 50 | 0 |
| Growth Value | 100 | 90 | 70 | 60 |
The task is to set up the initial simplex tableau that would determine how many kilograms of each food should be produced in order to maximize total growth.
- Let \( x_1 \) be the number of kilograms of food type P.
- Let \( x_2 \) be the number of kilograms of Q.
- Let \( x_3 \) be the number of kilograms of R.
- Let \( x_4 \) be the number of kilograms of S.
**Objective Function:**
\[ z = 100x_1 + 90x_2 + 70x_3 + 60x_4 \]
The goal is to maximize \( z \), the total growth value by determining optimal values for \( x_1, x_2, x_3, \) and \( x_4 \) within the constraints provided by the available nutrients A, B, and C.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F312de20a-253d-4366-a119-2f34c19d25d5%2F7ae6fa6b-b167-49cd-a863-03b0d3b8f07e%2F9lu3c0mn.jpeg&w=3840&q=75)
Transcribed Image Text:**Optimization Problem for Food Production**
A biologist has 600 kg of nutrient A, 800 kg of nutrient B, and 500 kg of nutrient C. These nutrients will be used to make 4 types of food, whose contents (in percent per kilogram of food) and whose 'growth values' are shown in the table below.
| | P | Q | R | S |
|-----|-----|-----|-----|-----|
| A | 0 | 0 | 12.5 | 37.5 |
| B | 0 | 20 | 37.5 | 62.5 |
| C | 100 | 80 | 50 | 0 |
| Growth Value | 100 | 90 | 70 | 60 |
The task is to set up the initial simplex tableau that would determine how many kilograms of each food should be produced in order to maximize total growth.
- Let \( x_1 \) be the number of kilograms of food type P.
- Let \( x_2 \) be the number of kilograms of Q.
- Let \( x_3 \) be the number of kilograms of R.
- Let \( x_4 \) be the number of kilograms of S.
**Objective Function:**
\[ z = 100x_1 + 90x_2 + 70x_3 + 60x_4 \]
The goal is to maximize \( z \), the total growth value by determining optimal values for \( x_1, x_2, x_3, \) and \( x_4 \) within the constraints provided by the available nutrients A, B, and C.
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