https://file.aiquickdraw.com/mathbot/file/17273964117202zncgylc.jpg

Let's go through the linear programming problem from the image step by step.

### Problem Overview:

1. **Vendor's Purchases:**
   - $ x $: kilograms of peanuts bought.
   - $ y $: kilograms of cashew nuts bought.

### Part (a):

#### (i) Inequalities for the minimum purchase:
- The vendor must buy a minimum of 10 kg of peanuts: $ x \geq 10 $.
- The vendor must buy a minimum of 5 kg of cashew nuts: $ y \geq 5 $.

#### (ii) Inequalities for the total quantity and cost:
- The total amount of nuts bought must not exceed 60 kg: $ x + y \leq 60 $.
- Peanuts cost $4 per kg and cashews cost $8 per kg. The vendor must spend at least $200:
  $ 4x + 8y \geq 200 $.

Thus, the four inequalities are:
1. $ x \geq 10 $
2. $ y \geq 5 $
3. $ x + y \leq 60 $
4. $ 4x + 8y \geq 200 $

### Part (b):

**Graphing the inequalities:**
- Use a scale of 2 cm to represent 10 kg on each axis.
- Draw the four inequalities on the graph and shade the region that satisfies all four inequalities.

### Part (c):

#### (i) Determining maximum profit:
- The profit from peanuts is $2 per kg, and from cashews, it is $5 per kg.
- Using the feasible region from the graph, find the values of $ x $ and $ y $ that maximize profit.
  
Profit function: $ P = 2x + 5y $.

#### (ii) Calculate the maximum profit based on the optimal $ x $ and $ y $ values from the graph.

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Would you like further explanation or any clarifications on this process? Here are some follow-up questions to consider:

1. How would you graph the inequality $ 4x + 8y \geq 200 $?
2. What strategies can be used to find the maximum profit in linear programming problems?
3. How does shading help identify the feasible region on a graph?
4. Can you describe how the inequalities create boundaries on the graph?
5. What impact does changing the price of peanuts or cashew nuts have on the profit?

**Tip:** When working on linear programming problems, always check the corner points of the feasible region for the best solution!

A vendor buys x kg of peanuts and y kg of cashew nuts. The problem asks for inequalities that satisfy certain purchase conditions, graphing these inequalities, and calculating the maximum profit based on the profit from selling peanuts and cashew nuts.

Learn how to solve a linear programming problem where a vendor buys peanuts and cashew nuts. Understand how to set up inequalities for minimum purchases, spending constraints, and use graphing techniques to find the optimal solution that maximizes profit.

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