https://file.aiquickdraw.com/mathbot/file/1727395354038denv8ex6.jpg

To assist you in drawing the graph for this question, let's first break down the inequalities based on the given conditions and then I'll guide you through plotting them.

### Part (a) Inequalities:
#### (i) Minimum purchase requirement:
1. She must buy a minimum of **10 kg of peanuts**:
   \[
   x \geq 10
   \]
2. She must buy a minimum of **5 kg of cashew nuts**:
   \[
   y \geq 5
   \]

#### (ii) Total nuts and cost constraints:
3. She buys no more than **60 kg of nuts** (total weight):
   \[
   x + y \leq 60
   \]
4. Total spending constraint: Peanuts cost $4 per kg and cashew nuts $8 per kg. The total cost is at least **$200**:
   \[
   4x + 8y \geq 200
   \]

### Step-by-Step Graphing Instructions:

- **Axes setup**: Let the horizontal axis represent $x$ (kilograms of peanuts) and the vertical axis represent $y$ (kilograms of cashew nuts).
- Use a scale of 2 cm = 10 kg on both axes (as suggested in the question).
  
### Plotting the inequalities:
1. **$x \geq 10$**: Draw a vertical line at $x = 10$. Shade to the right of this line (for $x \geq 10$).
   
2. **$y \geq 5$**: Draw a horizontal line at $y = 5$. Shade above this line (for $y \geq 5$).

3. **$x + y \leq 60$**: This is a straight line with slope -1 (since $y = 60 - x$). It intersects the $x$-axis at $x = 60$ and the $y$-axis at $y = 60$. Shade below this line.

4. **$4x + 8y \geq 200$**: This can be rewritten as $x + 2y \geq 50$. The line has slope -1/2 and intersects the $x$-axis at $x = 50$ and the $y$-axis at $y = 25$. Shade above this line.

### Conclusion:
The shaded region where all the inequalities overlap will be your solution region.

### Do you want me to generate this graph digitally, or do you need further details? 

Also, here are 5 relative questions for you to think about:
1. What happens to the feasible region if the minimum spending is increased?
2. How would the graph change if the price of peanuts increases to $6 per kg?
3. Can the vendor buy exactly 60 kg of nuts and meet all constraints?
4. What would happen if there was no minimum purchase requirement for cashew nuts?
5. How does the slope of each inequality impact the size of the solution region?

**Tip**: Make sure your graph is to scale to avoid inaccuracies in identifying the solution region.

A vendor buys x kg of peanuts and y kg of cashew nuts. (a) (i) To get a good bargain, she must buy a minimum of 10 kg of peanuts and a minimum of 5 kg of cashew nuts. Write TWO inequalities which satisfy these conditions. (ii) She buys no more than 60 kg of nuts. Peanuts cost $4.00 per kg and cashew nuts cost $8.00 per kg and she spends at least $200. Write TWO inequalities which satisfy these conditions. (b) Using a scale of 2 cm to represent 10 kg on each axis, draw the graph of the FOUR inequalities in (a)(i) and (a)(ii). On your graph, shade ONLY the region which satisfies all four inequalities. (c) The profit on the sale of 1 kg of peanuts is $2.00 and on 1 kg of cashew nuts is $5.00. (i) Using your graph, determine the number of kilograms of each type of nut the vendor must sell in order to make the maximum profit. (ii) Calculate the maximum profit.

Learn how to graph and solve linear inequalities based on a vendor's purchasing constraints for peanuts and cashew nuts. This problem includes finding the feasible region that satisfies the conditions and calculating maximum profit. Suitable for students in Grades 9-12.

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