8. One bag of "Fertilizer A" contains 2 units of phosphorus, 5 units of nitrogen, and 3 units of potassium. One bag of "Fertilizer B" contains 3 units of phosphorus, 4 units of nitrogen, and 6 units of potassium. One bag of "Fertilizer C" contains 5 units of phosphorus, 3 units of nitrogen and 4 units of potassium. A gardener needs to apply at least 75 units of phosphorus, at least 95 units of nitrogen, and at least 80 units of potassium. The price of one bag is $6 for Fertilizer A. $5 for Fertilizer B, and $8 for Fertilizer C. (a) formulate the problem and use the simplex method to determine how many bags of each fertilizer the gardener should purchase in order to meet the above requirements for the lowest possible price; note that the answers must be positive integers minimize subject to: = 6x₁ + 5+ ² 823 2+₁+34 1 523 * Fertilizer A-0 Write the answer in the box below 275 295 80 5 +₁1 44₂ 2 3x, 1642 423) X, , Xъ, X3 373. (b) what is that minimum total price? Write the answer in the box below RUBB 20 X2=22.7273 x3=1.3636 Fertilizer B=22 Fertilizer = 1 124.5455 (c) if all fertilizer is used and applied to the soil in the garden, then will excess amounts of any one of the essential substances (phosphorus, nitrogen, potassium) be applied? if so, which one(s) and in what amount(s)?
8. One bag of "Fertilizer A" contains 2 units of phosphorus, 5 units of nitrogen, and 3 units of potassium. One bag of "Fertilizer B" contains 3 units of phosphorus, 4 units of nitrogen, and 6 units of potassium. One bag of "Fertilizer C" contains 5 units of phosphorus, 3 units of nitrogen and 4 units of potassium. A gardener needs to apply at least 75 units of phosphorus, at least 95 units of nitrogen, and at least 80 units of potassium. The price of one bag is $6 for Fertilizer A. $5 for Fertilizer B, and $8 for Fertilizer C. (a) formulate the problem and use the simplex method to determine how many bags of each fertilizer the gardener should purchase in order to meet the above requirements for the lowest possible price; note that the answers must be positive integers minimize subject to: = 6x₁ + 5+ ² 823 2+₁+34 1 523 * Fertilizer A-0 Write the answer in the box below 275 295 80 5 +₁1 44₂ 2 3x, 1642 423) X, , Xъ, X3 373. (b) what is that minimum total price? Write the answer in the box below RUBB 20 X2=22.7273 x3=1.3636 Fertilizer B=22 Fertilizer = 1 124.5455 (c) if all fertilizer is used and applied to the soil in the garden, then will excess amounts of any one of the essential substances (phosphorus, nitrogen, potassium) be applied? if so, which one(s) and in what amount(s)?
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Please check my answers and need help on c
![**Problem 8: Fertilizer Optimization**
A gardener needs to determine the optimal combination of fertilizers to meet specific nutrient requirements for the lowest cost.
**Fertilizer Details:**
- **Fertilizer A**:
- 2 units of phosphorus
- 5 units of nitrogen
- 3 units of potassium
- Cost: $6 per bag
- **Fertilizer B**:
- 3 units of phosphorus
- 4 units of nitrogen
- 6 units of potassium
- Cost: $5 per bag
- **Fertilizer C**:
- 5 units of phosphorus
- 3 units of nitrogen
- 4 units of potassium
- Cost: $8 per bag
**Requirements:**
- At least 75 units of phosphorus
- At least 95 units of nitrogen
- At least 80 units of potassium
**Goal:**
Minimize the cost.
**Mathematical Formulation:**
- **Objective Function:**
\[
\text{Minimize } P = 6x_1 + 5x_2 + 8x_3
\]
where \(x_1\), \(x_2\), and \(x_3\) represent the number of bags of Fertilizers A, B, and C respectively.
- **Constraints:**
\[
\begin{align*}
2x_1 + 3x_2 + 5x_3 & \geq 75 \\
5x_1 + 4x_2 + 3x_3 & \geq 95 \\
3x_1 + 6x_2 + 4x_3 & \geq 80 \\
x_1, x_2, x_3 & \geq 0
\end{align*}
\]
**Solution:**
- Quantities:
- Fertilizer A: \( x_1 = 0 \)
- Fertilizer B: \( x_2 = 22.7273 \)
- Fertilizer C: \( x_3 = 1.3636 \)
- **Minimum Total Price:**
\[
124.5455
\]
**Excess Analysis:**
If all fertilizers are](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc62debf9-2696-4946-934a-f308bf93cf3b%2Fd705d1b1-b7bd-4639-8300-cb388f4b70a8%2Fz6wp5o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 8: Fertilizer Optimization**
A gardener needs to determine the optimal combination of fertilizers to meet specific nutrient requirements for the lowest cost.
**Fertilizer Details:**
- **Fertilizer A**:
- 2 units of phosphorus
- 5 units of nitrogen
- 3 units of potassium
- Cost: $6 per bag
- **Fertilizer B**:
- 3 units of phosphorus
- 4 units of nitrogen
- 6 units of potassium
- Cost: $5 per bag
- **Fertilizer C**:
- 5 units of phosphorus
- 3 units of nitrogen
- 4 units of potassium
- Cost: $8 per bag
**Requirements:**
- At least 75 units of phosphorus
- At least 95 units of nitrogen
- At least 80 units of potassium
**Goal:**
Minimize the cost.
**Mathematical Formulation:**
- **Objective Function:**
\[
\text{Minimize } P = 6x_1 + 5x_2 + 8x_3
\]
where \(x_1\), \(x_2\), and \(x_3\) represent the number of bags of Fertilizers A, B, and C respectively.
- **Constraints:**
\[
\begin{align*}
2x_1 + 3x_2 + 5x_3 & \geq 75 \\
5x_1 + 4x_2 + 3x_3 & \geq 95 \\
3x_1 + 6x_2 + 4x_3 & \geq 80 \\
x_1, x_2, x_3 & \geq 0
\end{align*}
\]
**Solution:**
- Quantities:
- Fertilizer A: \( x_1 = 0 \)
- Fertilizer B: \( x_2 = 22.7273 \)
- Fertilizer C: \( x_3 = 1.3636 \)
- **Minimum Total Price:**
\[
124.5455
\]
**Excess Analysis:**
If all fertilizers are
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