Roman Buys Seed Packets For The School Garden. One Packet Of Radish Seeds Costs $ 1.75 \$1.75 $1.75 . One Packet Of Tomato Seeds Costs $ 2.50 \$2.50 $2.50 . Let R R R Represent The Number Of Radish Seeds, And Let T T T Represent The Number Of

Introduction

As a student, Roman is involved in maintaining the school garden. Recently, he went to the gardening store to purchase seed packets for the garden. The store had two types of seeds available: radish seeds and tomato seeds. Roman wanted to buy the right number of each type of seed to ensure the garden had a diverse and thriving collection of plants. In this article, we will explore the math problem that Roman faced when selecting the seed packets.

The Cost of Radish Seeds

One packet of radish seeds costs $\$1.75$. This is a fixed cost, and Roman will have to pay this amount regardless of the number of packets he buys. Let's represent the number of radish seeds as $r$. The cost of $r$ packets of radish seeds can be calculated as $1.75r$.

The Cost of Tomato Seeds

One packet of tomato seeds costs $\$2.50$. Similar to the radish seeds, this is a fixed cost, and Roman will have to pay this amount regardless of the number of packets he buys. Let's represent the number of tomato seeds as $t$. The cost of $t$ packets of tomato seeds can be calculated as $2.50t$.

The Total Cost

The total cost of the seed packets can be calculated by adding the cost of the radish seeds and the cost of the tomato seeds. This can be represented as:

$$Total\ Cost = 1.75r + 2.50t$$

The Budget Constraint

Roman has a limited budget for the seed packets. Let's assume that his budget is $B$. The total cost of the seed packets cannot exceed this budget. This can be represented as:

$$1.75r + 2.50t \leq B$$

The Objective

The objective of Roman is to select the right number of radish seeds and tomato seeds such that the total cost is minimized. This can be represented as:

$$Minimize: 1.75r + 2.50t$$

Subject to:

• $1.75r + 2.50t \leq B$
• $r \geq 0$
• $t \geq 0$

Solving the Problem

To solve this problem, we can use linear programming techniques. One approach is to use the graphical method. We can plot the budget constraint on a graph and find the feasible region. The optimal solution will be the point in the feasible region that minimizes the total cost.

Graphical Method

Let's assume that Roman's budget is $B = 50$. We can plot the budget constraint on a graph as follows:

Introduction

In our previous article, we explored the math problem that Roman faced when selecting the seed packets for the school garden. We used linear programming techniques to find the optimal solution. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the objective of Roman's problem?

A: The objective of Roman's problem is to select the right number of radish seeds and tomato seeds such that the total cost is minimized.

Q: What is the budget constraint?

A: The budget constraint is that the total cost of the seed packets cannot exceed Roman's budget, which is $50.

Q: How can we represent the budget constraint mathematically?

A: The budget constraint can be represented mathematically as:

$$1.75r + 2.50t \leq 50$$

Q: What is the feasible region?

A: The feasible region is the set of all possible solutions that satisfy the budget constraint. It is the area on the graph where the total cost is less than or equal to $50.

Q: How can we find the optimal solution?

A: The optimal solution can be found by identifying the point in the feasible region that minimizes the total cost. This can be done using linear programming techniques, such as the graphical method.

Q: What is the optimal solution?

A: The optimal solution is the point in the feasible region where the total cost is minimized. In this case, the optimal solution is:

$$r = 20, t = 10$$

Q: What is the minimum total cost?

A: The minimum total cost is $35, which occurs when $r = 20$ and $t = 10$.

Q: What is the significance of the optimal solution?

A: The optimal solution is significant because it represents the most cost-effective way for Roman to select the seed packets. By choosing $r = 20$ and $t = 10$, Roman can minimize the total cost while still satisfying the budget constraint.

Q: Can we use other methods to solve the problem?

A: Yes, we can use other methods to solve the problem, such as the simplex method or the dual simplex method. These methods can be more efficient than the graphical method for larger problems.

Q: How can we extend the problem to include other variables?

A: We can extend the problem to include other variables, such as the number of packets of lettuce seeds or the number of packets of cucumber seeds. This would require modifying the budget constraint and the objective function to include the new variables.

Conclusion

In this article, we answered some frequently asked questions related to Roman's problem of selecting the seed packets for the school garden. We used linear programming techniques to find the optimal solution and discussed the significance of the optimal solution. We also explored the possibility of extending the problem to include other variables.