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The question asks to select the most appropriate domain for the profit function of a bakery, given by the equation \( p(x) = -0.2(x - 60)^2 + 150 \), where \( x \) represents the number of pastries sold per day. The domain should represent realistic values for \( x \).

Here are the options to evaluate:
1. **\( x \) is an integer, such that \( x \geq 0 \)**: This is practical as you can't sell a fractional pastry and sales cannot be negative.
2. **\( x \) is a rational number**: This allows for fractional pastries, which isn't realistic.
3. **\( \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \} \)**: This includes negative numbers, which aren't viable for the number of pastries sold.
4. **\( x \) is a real number, such that \( [0, \infty) \)**: This includes all non-negative real numbers, which is not practical because it allows for fractional pastries.
5. **\( x \) is an irrational number**: This option is not practical since selling an irrational number of pastries is not possible.

The best choice here is:
1. **\( x \) is an integer, such that \( x \geq 0 \)**, as it best fits the realistic scenario of counting whole pastries.

Would you like further details on how this function works or have other questions about the topic?

Here are five additional related questions you might find interesting:
1. What is the maximum profit the bakery can achieve according to the function \( p(x) \)?
2. How many pastries must the bakery sell to achieve maximum profit?
3. How does the shape of the parabola \( -0.2(x - 60)^2 + 150 \) relate to the bakery's profits?
4. Can the profit ever become negative according to this model? Under what circumstances?
5. How would the profit function change if the bakery's base cost per pastry or sale price were to change?

**Tip:** Always check the practical implications of mathematical models in real-life scenarios to ensure they reflect possible and logical outcomes.

85°C Bakery sells more pastries when it reduces the prices. The profit is modeled by the function p(x) = -0.2(x - 60)^2 + 150, where x is the number of pastries sold per day. Which domain would be the most appropriate for this function?

Explore how to find the most appropriate domain for the profit function p(x) = -0.2(x - 60)^2 + 150, where x represents the number of pastries sold per day. This question involves algebra, quadratic functions, and real-world applications, suitable for students in Grades 8-10.

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