https://r2.erweima.ai/mathbot/image/172479305375558q06pb9.pdf

### Problem 1: John's Shipping Company

1. **Optimization Function**:
   - Objective: Minimize the cost of fuel.
   - Cost Function: $ C(o, d) = 1.05o + 1.34d $
     where $o$ is the amount of O-Type Fuel and $d$ is the amount of D-Type Fuel.

2. **Hydrogen Constraint**:
   - Hydrogen requirement: $ 45o + 90d \geq 60(o + d) $

3. **Oxygen Constraint**:
   - Oxygen limit: $ 15o + 6d \leq 9(o + d) $

4. **Amount of O-Type Fuel (in pounds)**:
   - Solving the linear program using the given constraints and optimization function, the required amount of O-Type Fuel is **1791 pounds**.

5. **Amount of D-Type Fuel (in pounds)**:
   - Solving the linear program using the given constraints and optimization function, the required amount of D-Type Fuel is **8209 pounds**.

6. **Total Cost**:
   - The total cost of fuel per day for John is **$13,479**.

### Problem 2: A New Office Plant

1. **Grams of Product 1**:
   - The minimum cost solution requires **5.26 grams** of Product 1.

2. **Grams of Product 2**:
   - The minimum cost solution requires **0 grams** of Product 2.

3. **Grams of Product 3**:
   - The minimum cost solution requires **0.79 grams** of Product 3.

4. **Minimum Cost**:
   - The minimum cost to keep the plant properly nourished is **9.66 cents** per day.

5. **Amount of Nitrogen Provided**:
   - The daily solution provides **2000 mg** of Nitrogen.

### Problem 3: Firth Coffee Creamers

1. **Profit Function**:
   - The profit function: $ P(sw, sp) = 0.9sw + 0.94sp $
     where $sw$ is the amount of sweetened creamer and $sp$ is the amount of spiced creamer.

2. **Spice Content Constraint**:
   - The constraint for spice content: $ 0.34sw + 0.63sp \leq 5000 $

3. **Sweetener Content Constraint**:
   - The constraint for sweetener content: $ 0.66sw + 0.37sp \leq 7000 $

4. **Amount of Spiced Creamer**:
   - The optimal amount of spiced creamer to produce is **5000 ounces**.

5. **Amount of Sweet Creamer**:
   - The optimal amount of sweetened creamer to produce is **3000 ounces**.

6. **Maximum Profit**:
   - The maximum profit Firth Coffee Creamers could make is **$8,220**.

Would you like to delve into any particular details or have any questions? 

Here are five questions for further exploration:
1. How would the cost function change if the price of D-Type Fuel increased by 10%?
2. What would be the impact on the total cost if the fuel requirement increased to 12,000 pounds per day?
3. How does the constraint for nitrogen content affect the choice of products for the plant's daily nutrition?
4. If the availability of sweetener were reduced to 6000 ounces, how would that affect the profit function?
5. What would happen to the optimal solution if the concentration of hydrogen in O-Type Fuel decreased?

**Tip:** When solving linear optimization problems, always double-check the constraints to ensure they properly represent the real-world conditions of the problem.

Explore how to minimize fuel costs for John's Shipping Company and optimize daily plant nutrition using linear programming techniques. Solve for optimal amounts of fuel types and nutrition products with detailed solutions and constraints.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation