The height (in feet) of a punted football is a function of the time(in seconds) the ball is in the air. the function is defined by h(t)=-7t^2+48t. what is likely the domain of the function?a. all real numbers b. negative rational numbers c. only whole numbers d. positive real numbers

The function given is:

\[
h(t) = -7t^2 + 48t
\]

This represents the height \( h(t) \) of a punted football in feet as a function of time \( t \) in seconds. 

### Understanding the Domain

1. **Nature of the variables**:  
   - The variable \( t \) represents time, which cannot be negative. Thus, negative values are not valid in this context.
   - The function also involves squaring \( t \) and multiplying by constants, which are defined for all real numbers, but only non-negative values make sense for time.

2. **Height context**:  
   - The football is in the air only for a limited duration of time. After that, the height will be 0 when it hits the ground again.

### Determining the Valid Domain

Given that time \( t \) cannot be negative, we exclude all options involving negative numbers or arbitrary real numbers.

- **Option (a) "all real numbers"** is not correct since negative times are not meaningful.
- **Option (b) "negative rational numbers"** is also incorrect since time can't be negative.
- **Option (c) "only whole numbers"** might suggest time is measured in whole seconds only, but time can be any real positive value.
- **Option (d) "positive real numbers"** is correct because it considers all positive values for time, which are meaningful for this function. However, technically, the function's domain would also include zero (starting time).

### Conclusion

**The correct answer is (d) positive real numbers.**

If you have any questions or would like more details, feel free to ask!

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**Here are 5 related questions to consider:**

1. How would you find the maximum height of the punted football using the given function?
2. What is the time when the football reaches its maximum height?
3. How would you determine the total time the football is in the air?
4. Can you determine the height of the football after 3 seconds using the function?
5. What type of function is this, and what is the significance of its coefficients?

**Tip:** When analyzing functions related to real-world scenarios, always think about the physical constraints and context to determine the correct domain and range!

Explore the domain of the quadratic function h(t) = -7t^2 + 48t, which represents the height of a punted football over time. Learn why the domain consists of positive real numbers, essential for understanding the function's behavior in real-world scenarios. Suitable for advanced high school students.

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