**Tasty Yogurt Manufacturing Co. makes 10 cents per container of plain yogurt that it sells and 12 cents per container of vanilla yogurt.  They want to make a monthly profit of at least $3,200.  This month, Tasty expects to sell at least 12,000 cups of plain yogurt (*p*), but no more than 15,000 cups of vanilla yogurt (*v*).**

**Part A:**

**Which of the following inequalities represent this situation?  Select *all* that apply.**

A. 10*p* +12*v* ≥ 3200

B. 0.10*p* + 0.12*v* ≥ 3200

C. *p* ≥ 12,000

D. *p* ≤ 12,000

E. *v* ≥15,000

F. *v* ≤ 15,000

To solve this problem, we need to translate the information given into mathematical inequalities that represent the situation. Let's go through each part step by step.

### 1. **Profit Condition**
Tasty Yogurt Manufacturing Co. wants to make a monthly profit of at least $3,200. The profit per container is:

- 10 cents per container of plain yogurt (which is $0.10p$ in dollars)
- 12 cents per container of vanilla yogurt (which is $0.12v$ in dollars)

The total profit is given by:
\[
\text{Total Profit} = 0.10p + 0.12v
\]

Since they want the total profit to be at least $3,200, the inequality is:
\[
0.10p + 0.12v \geq 3200
\]

This inequality is represented by **Option B**.

**Note:** **Option A** is incorrect because it uses 10 and 12 instead of 0.10 and 0.12, which would result in a much higher required profit than intended.

### 2. **Plain Yogurt Sales**
Tasty expects to sell at least 12,000 cups of plain yogurt. This can be written as:
\[
p \geq 12,000
\]

This inequality is represented by **Option C**.

**Note:** **Option D** is incorrect because it suggests $ p \leq 12,000 $, which contradicts the problem statement.

### 3. **Vanilla Yogurt Sales**
Tasty expects to sell no more than 15,000 cups of vanilla yogurt. This can be written as:
\[
v \leq 15,000
\]

This inequality is represented by **Option F**.

**Note:** **Option E** is incorrect because it suggests $ v \geq 15,000 $, which contradicts the problem statement.

### Conclusion
The correct inequalities that represent the situation are:

- **B. $0.10p + 0.12v \geq 3200$**
- **C. $p \geq 12,000$**
- **F. $v \leq 15,000$**

Would you like more details or have any questions?

Here are five related questions for further practice:

1. If Tasty sells exactly 15,000 cups of vanilla yogurt, how many plain yogurt cups must they sell to meet the $3,200 profit goal?
2. What is the minimum number of vanilla yogurt cups needed if Tasty sells exactly 12,000 plain yogurt cups?
3. How would the inequalities change if the profit target was increased to $4,000?
4. If Tasty sells 13,000 plain yogurt cups and 14,000 vanilla yogurt cups, what will be their total profit?
5. How does the maximum number of vanilla yogurt cups affect the minimum number of plain yogurt cups required to meet the profit goal?

**Tip:** Always convert units correctly when dealing with profit calculations to avoid errors in inequalities and equations.

Learn how Tasty Yogurt Manufacturing Co. uses mathematical inequalities to maximize profit from sales of plain and vanilla yogurt cups. Solve for the minimum sales requirements to achieve a monthly profit goal of $3,200. Suitable for Grades 10-12.

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