A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen, and 10 pounds of potash. The minimum monthly requirements are 450 pounds of phosphoric acid, 990 pounds of nitrogen, and 210 pounds of potash. a. If mix A costs $20 per cubic yard and mix B costs $30 per cubic yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimum cost? b. What is this cost? ### Optimizing Plant Food Blends for Nutrient Requirements**Problem Statement:**A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains:- 20 pounds of phosphoric acid- 30 pounds of nitrogen- 5 pounds of potashEach cubic yard of mix B contains:- 10 pounds of phosphoric acid- 30 pounds of nitrogen- 10 pounds of potashThe minimum monthly requirements for the nutrients are as follows:- 450 pounds of phosphoric acid- 990 pounds of nitrogen- 210 pounds of potash**Pricing:**- Mix A costs $20 per cubic yard- Mix B costs $30 per cubic yard**Objective:**Determine how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimum cost and calculate this cost.---**Solution:**To solve this, we need to use a mathematical optimization approach. Let's set up the equation with the variables representing the quantity of each mix:1. Let $ x $ be the number of cubic yards of mix A.2. Let $ y $ be the number of cubic yards of mix B.The nutrient requirements can be written as:- Phosphoric acid: $ 20x + 10y \geq 450 $- Nitrogen: $ 30x + 30y \geq 990 $- Potash: $ 5x + 10y \geq 210 $The cost equation that we want to minimize is:\[ \text{Cost} = 20x + 30y \]**Constraints:**1. $ 20x + 10y \geq 450 $2. $ 30x + 30y \geq 990 $3. $ 5x + 10y \geq 210 $**Blended Quantities and Cost:**According to the problem solution:\[ \text{The farmer should blend } \boxed{x} \text{ yd}^3 \text{ of mix A and } \boxed{y} \text{ yd}^3 \text{ of mix B to meet the minimum monthly requirements at a minimum cost.} \]---This problem requires solving a system of linear inequalities to find the optimal values for $x$ and \(y

x=yd3 of mix A y=yd3 of mix B make a table from the given data

A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen, and 10 pounds of potash. The minimum monthly requirements are 450 pounds of phosphoric acid, 990 pounds of nitrogen, and 210 pounds of potash. If mix A costs $20 per cubic yard and mix B costs $30 per cubic yard, how many cubic yards of each mix should the farmer blend meet the minimum monthly requirements at a minimum cost? What is this cost? If mix A costs $20 per cubic yard and mix B costs $30 per cubic yard, the farmer should blend O ya of mix A andOya of mix B to meet the minimum monthly requirements at a minimum cost.

A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen, and 10 pounds of potash. The minimum monthly requirements are 450 pounds of phosphoric acid, 990 pounds of nitrogen, and 210 pounds of potash. If mix A costs $20 per cubic yard and mix B costs $30 per cubic yard, how many cubic yards of each mix should the farmer blend meet the minimum monthly requirements at a minimum cost? What is this cost? If mix A costs $20 per cubic yard and mix B costs $30 per cubic yard, the farmer should blend O ya of mix A andOya of mix B to meet the minimum monthly requirements at a minimum cost.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.4: Linear Programming

Problem 23E

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Question

a. If mix A costs $20 per cubic yard and mix B costs $30 per cubic yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimum cost? b. What is this cost?

### Optimizing Plant Food Blends for Nutrient Requirements

**Problem Statement:**

A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains:

- 20 pounds of phosphoric acid
- 30 pounds of nitrogen
- 5 pounds of potash

Each cubic yard of mix B contains:

- 10 pounds of phosphoric acid
- 30 pounds of nitrogen
- 10 pounds of potash

The minimum monthly requirements for the nutrients are as follows:

- 450 pounds of phosphoric acid
- 990 pounds of nitrogen
- 210 pounds of potash

**Pricing:**

- Mix A costs $20 per cubic yard
- Mix B costs $30 per cubic yard

**Objective:**

Determine how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimum cost and calculate this cost.

---

**Solution:**

To solve this, we need to use a mathematical optimization approach. Let's set up the equation with the variables representing the quantity of each mix:

1. Let $ x $ be the number of cubic yards of mix A.
2. Let $ y $ be the number of cubic yards of mix B.

The nutrient requirements can be written as:

- Phosphoric acid: $ 20x + 10y \geq 450 $
- Nitrogen: $ 30x + 30y \geq 990 $
- Potash: $ 5x + 10y \geq 210 $

The cost equation that we want to minimize is:

\[ \text{Cost} = 20x + 30y \]

**Constraints:**

1. $ 20x + 10y \geq 450 $
2. $ 30x + 30y \geq 990 $
3. $ 5x + 10y \geq 210 $

**Blended Quantities and Cost:**

According to the problem solution:

\[ \text{The farmer should blend } \boxed{x} \text{ yd}^3 \text{ of mix A and } \boxed{y} \text{ yd}^3 \text{ of mix B to meet the minimum monthly requirements at a minimum cost.} \]

---

This problem requires solving a system of linear inequalities to find the optimal values for $x$ and \(y

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