An Object Is Dropped From A Height Of 400 Feet. Its Height, $h(t$\] In Feet, After $t$ Seconds, Is Modeled By The Equation $h(t) = 400 - 16t^2$.What Is The Domain Of $h(t$\]?A. All Real Numbers Greater Than 0 B. All

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In this article, we will explore the domain of a quadratic function that models the height of an object dropped from a height of 400 feet. The function is given by the equation $h(t) = 400 - 16t^2$, where $t$ represents time in seconds and $h(t)$ represents the height of the object in feet.

The Quadratic Function

The given quadratic function is $h(t) = 400 - 16t^2$. This function represents the height of the object at any given time $t$. The coefficient of the squared term, $-16$, is negative, indicating that the parabola opens downward. This means that the function will have a maximum value, which occurs at the vertex of the parabola.

Finding the Domain

To find the domain of the function, we need to determine the set of all possible input values for which the function is defined. In this case, the function is defined for all real numbers $t$. However, we need to consider the physical context of the problem. Since the object is dropped from a height of 400 feet, the height of the object cannot be negative. Therefore, we need to find the values of $t$ for which $h(t) \geq 0$.

Solving the Inequality

To solve the inequality $h(t) \geq 0$, we need to substitute the expression for $h(t)$ and simplify:

$$400 - 16t^2 \geq 0$$

Subtracting 400 from both sides gives:

$$-16t^2 \geq -400$$

Dividing both sides by $-16$ gives:

$$t^2 \leq 25$$

Taking the square root of both sides gives:

$$|t| \leq 5$$

This means that $t$ must be between $-5$ and $5$, inclusive.

Conclusion

In conclusion, the domain of the function $h(t) = 400 - 16t^2$ is the set of all real numbers $t$ such that $-5 \leq t \leq 5$. This means that the function is defined for all real numbers between $-5$ and $5$, inclusive.

Answer

The correct answer is:

• B. all real numbers between -5 and 5, inclusive

Final Thoughts

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Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined.

Q: What is the significance of the domain of a function?

A: The domain of a function is important because it determines the set of all possible input values for which the function is defined. This is crucial in understanding the behavior and properties of the function.

Q: How do you find the domain of a function?

A: To find the domain of a function, you need to determine the set of all possible input values for which the function is defined. This can be done by analyzing the function's equation, identifying any restrictions or limitations, and determining the set of all possible input values that satisfy these conditions.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values that the function can produce.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. This occurs when there are no input values for which the function is defined.

Q: Can the domain of a function be infinite?

A: Yes, the domain of a function can be infinite. This occurs when there are an infinite number of input values for which the function is defined.

Q: How do you determine the domain of a quadratic function?

A: To determine the domain of a quadratic function, you need to analyze the function's equation and identify any restrictions or limitations. This can be done by examining the function's graph, identifying any vertical asymptotes or holes, and determining the set of all possible input values that satisfy these conditions.

Q: What is the significance of the domain of a quadratic function?

A: The domain of a quadratic function is significant because it determines the set of all possible input values for which the function is defined. This is crucial in understanding the behavior and properties of the function.

Q: Can the domain of a quadratic function be restricted?

A: Yes, the domain of a quadratic function can be restricted. This occurs when there are input values for which the function is not defined.

Q: How do you restrict the domain of a quadratic function?

A: To restrict the domain of a quadratic function, you need to identify any input values for which the function is not defined and exclude them from the domain.

Q: What is the difference between a restricted and unrestricted domain?

A: A restricted domain is a domain that has been limited or restricted in some way, while an unrestricted domain is a domain that has no limitations or restrictions.

Q: Can the domain of a function be changed?

A: Yes, the domain of a function can be changed. This can be done by modifying the function's equation or by restricting the domain in some way.

Q: How do you change the domain of a function?

A: To change the domain of a function, you need to modify the function's equation or restrict the domain in some way. This can be done by analyzing the function's graph, identifying any vertical asymptotes or holes, and determining the set of all possible input values that satisfy these conditions.

Q: What is the significance of changing the domain of a function?

A: Changing the domain of a function can have significant implications for the function's behavior and properties. It can affect the function's graph, its range, and its overall behavior.

Q: Can changing the domain of a function affect the function's range?

A: Yes, changing the domain of a function can affect the function's range. This occurs when the domain is restricted in some way, which can limit the function's output values.

Q: How do you determine the range of a function?

A: To determine the range of a function, you need to analyze the function's equation and identify any output values that the function can produce. This can be done by examining the function's graph, identifying any horizontal asymptotes or holes, and determining the set of all possible output values that satisfy these conditions.