Working with chemists at Virginia Tech and GeorgeWashington Universities, landscape contractorKenneth Golding blended his own fertilizer, called“Golding-Grow.” It consists of four chemical com-pounds: C-30, C-92, D-21, and E-11. The cost perpound for each compound is indicated as follows:CHEMICAL COMPOUND COST PER POUND ($)C-30 0.12C-92 0.09D-21 0.11E-11 0.04The specifications for Golding-Grow are as follows:(1) E-11 must constitute at least 15% of the blend; (2)C-92 and C-30 must together constitute at least 45%of the blend; (3) D-21 and C-92 can together consti-tute no more than 30% of the blend; and (4) Golding-Grow is packaged and sold in 50-pound bag

This is an optimization problem involving a mixture of chemical compounds. The goal is likely to minimize the cost of producing a 50-pound bag of the "Golding-Grow" fertilizer, subject to the given constraints.

We can break this down as follows:

### Variables:
Let the amounts of each compound in the 50-pound bag be represented by:
- $ x_1 $ = pounds of C-30 in the mix
- $ x_2 $ = pounds of C-92 in the mix
- $ x_3 $ = pounds of D-21 in the mix
- $ x_4 $ = pounds of E-11 in the mix

### Objective:
Minimize the total cost of the blend. The total cost of the mixture is:
\[
\text{Total cost} = 0.12x_1 + 0.09x_2 + 0.11x_3 + 0.04x_4
\]

### Constraints:
1. The total weight of the mixture must be 50 pounds:
   \[
   x_1 + x_2 + x_3 + x_4 = 50
   \]

2. E-11 must constitute at least 15% of the blend:
   \[
   x_4 \geq 0.15 \times 50 = 7.5
   \]

3. C-92 and C-30 must together constitute at least 45% of the blend:
   \[
   x_1 + x_2 \geq 0.45 \times 50 = 22.5
   \]

4. D-21 and C-92 together can constitute no more than 30% of the blend:
   \[
   x_3 + x_2 \leq 0.30 \times 50 = 15
   \]

5. Each of the variables must be non-negative:
   \[
   x_1, x_2, x_3, x_4 \geq 0
   \]

### Optimization Problem:
Minimize the cost function:
\[
0.12x_1 + 0.09x_2 + 0.11x_3 + 0.04x_4
\]
Subject to the constraints:
\[
\begin{aligned}
x_1 + x_2 + x_3 + x_4 &= 50 \\
x_4 &\geq 7.5 \\
x_1 + x_2 &\geq 22.5 \\
x_3 + x_2 &\leq 15 \\
x_1, x_2, x_3, x_4 &\geq 0
\end{aligned}
\]

To solve this, we can use methods such as linear programming.

Would you like to proceed with solving this optimization using a method like the Simplex algorithm, or would you like further clarification on any part of this setup?

### 5 Related Questions:
1. How can we formulate this problem as a linear program in matrix form?
2. What are some methods to solve linear programming problems, such as the Simplex method?
3. What would happen if some of the constraints were relaxed or tightened?
4. How does the choice of chemical compound proportions affect the overall cost?
5. How can sensitivity analysis be applied to assess the impact of changes in chemical costs?

**Tip:** When working with optimization problems, always check if constraints can be combined or simplified to reduce complexity before solving.

Working with chemists at Virginia Tech and George Washington Universities, landscape contractor Kenneth Golding blended his own fertilizer, called 'Golding-Grow.' It consists of four chemical compounds: C-30, C-92, D-21, and E-11. The cost per pound for each compound is indicated as follows: C-30 = $0.12, C-92 = $0.09, D-21 = $0.11, and E-11 = $0.04. The specifications for Golding-Grow are: (1) E-11 must constitute at least 15% of the blend; (2) C-92 and C-30 must together constitute at least 45% of the blend; (3) D-21 and C-92 together can constitute no more than 30% of the blend; and (4) Golding-Grow is packaged and sold in 50-pound bags.

Learn how to solve the optimization problem involving the Golding-Grow fertilizer mix using linear programming. This problem covers key concepts such as cost minimization, constraints on chemical compound proportions, and linear inequalities. Suitable for university-level students.

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