https://file.aiquickdraw.com/mathbot/file/1726099015724kpjopqvj.jpg

Let's analyze and solve each part of the problem shown in the image:

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**(1)** The function \( f(t) \) represents the number of people eating in a restaurant \( t \) minutes after 5 PM.

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### Part a)
You are given \( f(4) = 21 \). This means that 4 minutes after 5 PM, there are 21 people eating. 

- **Correct answer**: **C** There are 21 people eating at 5:04 PM.

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### Part b)
You are given \( f(a) = 38 \). This means that at some time \( a \) minutes after 5 PM, there are 38 people eating.

- **Correct answer**: **A** \( a \) minutes after 5 PM, there are 38 people eating.

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### Part c)
You are given \( f(38) = b \). This means that 38 minutes after 5 PM, there are \( b \) people eating. We don't know what \( b \) is, but we know that it represents the number of people eating 38 minutes after 5 PM.

- **Correct answer**: **E** None of the above (since none of the options precisely describe the significance).

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### Part d)
You are given the expression \( n = f(t) \). This simply means that \( n \), the number of people eating, depends on \( t \), the time after 5 PM.

- **Correct answer**: **C** \( n \) minutes after 5 PM, there are \( t \) people eating.

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Let me know if you would like additional details or clarifications on any part.

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**Related Questions:**
1. How does a function like \( f(t) \) change when the number of people increases every minute?
2. What are examples of other real-world scenarios that can be modeled with functions like \( f(t) \)?
3. How would the interpretation of \( f(t) \) change if the function was nonlinear?
4. What does it mean graphically when we say \( f(4) = 21 \)?
5. How would we solve for \( t \) if we were given \( f(t) = 50 \)?

**Tip:** Always pay close attention to what the function's input and output represent (e.g., time and number of people) to better interpret function values.

Let f(t) denote the number of people eating in a restaurant t minutes after 5 PM. Answer the following questions:

 a) Which of the following statements best describes the significance of the expression f(4) = 21?

 b) Which of the following statements best describes the significance of the expression f(a) = 38?

 c) Which of the following statements best describes the significance of the expression f(38) = b?

 d) Which of the following statements best describes the significance of the expression n = f(t)?

Explore how to interpret function notation in a real-world scenario where f(t) represents the number of people eating in a restaurant t minutes after 5 PM. This problem involves analyzing statements about time and quantities, ideal for students in Grades 6-9.

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