To determine how many cups of vanilla yogurt (v) Tasty Yogurt Manufacturing Company needs to sell when they are selling 12,000 cups of plain yogurt (p) to achieve their profit goal of at least $3,200, we can set up the equation based on the profits from both types of yogurt.
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Profit from plain yogurt (p):
- Profit per container of plain yogurt = 19 cents = $0.19
- If they sell 12,000 cups of plain yogurt: \[ \text{Profit from plain yogurt} = 12,000 \times 0.19 = 2,280 \text{ dollars} \]
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Profit from vanilla yogurt (v):
- Profit per container of vanilla yogurt = 12 cents = $0.12
- If they sell \( v \) cups of vanilla yogurt: \[ \text{Profit from vanilla yogurt} = v \times 0.12 \]
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Total profit must be at least $3,200: \[ \text{Total profit} = \text{Profit from plain yogurt} + \text{Profit from vanilla yogurt} \] \[ 2,280 + 0.12v \geq 3,200 \]
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Solving the inequality: \[ 0.12v \geq 3,200 - 2,280 \] \[ 0.12v \geq 920 \] \[ v \geq \frac{920}{0.12} \] \[ v \geq 7,666.67 \]
Since the company cannot sell a fraction of a container, they will need to sell at least 7,667 cups of vanilla yogurt.
Conclusion: If Tasty Yogurt Manufacturing Company sells 12,000 cups of plain yogurt, they need to sell at least 7,667 cups of vanilla yogurt to meet their profit goal of at least $3,200.
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