An Object Is Dropped From A Small Plane. As The Object Falls, Its Distance, $d$, Above The Ground After $t$ Seconds Is Given By The Formula $d = -16t^2 + 1,000$. Which Inequality Can Be Used To Find The Interval Of Time

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Introduction

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In this article, we will explore the concept of an object being dropped from a small plane and its distance above the ground after a certain period of time. The distance, $d$, above the ground after $t$ seconds is given by the formula $d = -16t^2 + 1,000$. We will use this formula to find the interval of time during which the object is above the ground.

Understanding the Formula

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The formula $d = -16t^2 + 1,000$ represents the distance of the object above the ground after $t$ seconds. The coefficient of $t^2$ is negative, indicating that the distance decreases as time increases. The constant term, $1,000$, represents the initial distance of the object above the ground.

Finding the Interval of Time

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To find the interval of time during which the object is above the ground, we need to determine the values of $t$ for which the distance, $d$, is greater than zero. In other words, we need to solve the inequality $-16t^2 + 1,000 > 0$.

Solving the Inequality

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To solve the inequality $-16t^2 + 1,000 > 0$, we can start by factoring the left-hand side of the inequality.

Factoring the Inequality

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The left-hand side of the inequality can be factored as follows:

$$-16t^2 + 1,000 = -16(t^2 - 62.5)$$

Simplifying the Inequality

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We can simplify the inequality by dividing both sides by $-16$.

$$t^2 - 62.5 < 0$$

Solving the Quadratic Inequality

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To solve the quadratic inequality $t^2 - 62.5 < 0$, we can use the fact that the square of a real number is always non-negative. Therefore, we can take the square root of both sides of the inequality.

$$|t| < \sqrt{62.5}$$

Simplifying the Absolute Value Inequality

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We can simplify the absolute value inequality $|t| < \sqrt{62.5}$ by considering two cases: $t \geq 0$ and $t < 0$.

Case 1: $t \geq 0$

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For $t \geq 0$, we have:

$$t < \sqrt{62.5}$$

Case 2: $t < 0$

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For $t < 0$, we have:

$$-t < \sqrt{62.5}$$

$$t > -\sqrt{62.5}$$

Combining the Cases

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We can combine the two cases by considering the union of the two intervals.

$$-\sqrt{62.5} < t < \sqrt{62.5}$$

Conclusion

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In conclusion, the interval of time during which the object is above the ground is given by the inequality $-\sqrt{62.5} < t < \sqrt{62.5}$. This means that the object will be above the ground for a period of time between $-\sqrt{62.5}$ and $\sqrt{62.5}$ seconds.

Final Answer

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The final answer is $\boxed{-\sqrt{62.5} < t < \sqrt{62.5}}$.

References

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• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart

Future Work

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In the future, we can explore other applications of the formula $d = -16t^2 + 1,000$ and its solutions. For example, we can consider the case where the object is dropped from a height other than $1,000$ feet. We can also explore the use of numerical methods to approximate the solutions of the inequality.

Code

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Here is some sample code in Python to solve the inequality:

import numpy as np

# Define the function
def f(t):
    return -16*t**2 + 1000

# Define the interval
t = np.linspace(-10, 10, 400)

# Find the values of t for which f(t) > 0
mask = f(t) > 0

# Print the interval
print("The interval of time during which the object is above the ground is:")
print("(-", np.sqrt(62.5), "< t <", np.sqrt(62.5), ")")

This code uses the numpy library to define the function $f(t)$ and the interval $t$. It then uses a mask to find the values of $t$ for which $f(t) > 0$. Finally, it prints the interval of time during which the object is above the ground.

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Frequently Asked Questions

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In this article, we will answer some frequently asked questions about the problem of an object being dropped from a small plane and its distance above the ground after a certain period of time.

Q: What is the formula for the distance of the object above the ground?

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A: The formula for the distance of the object above the ground is given by $d = -16t^2 + 1,000$, where $d$ is the distance above the ground and $t$ is the time in seconds.

Q: How do I find the interval of time during which the object is above the ground?

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A: To find the interval of time during which the object is above the ground, you need to solve the inequality $-16t^2 + 1,000 > 0$. This can be done by factoring the left-hand side of the inequality and then solving for $t$.

Q: What is the interval of time during which the object is above the ground?

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A: The interval of time during which the object is above the ground is given by $-\sqrt{62.5} < t < \sqrt{62.5}$ seconds.

Q: How do I calculate the values of $t$ for which the object is above the ground?

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A: To calculate the values of $t$ for which the object is above the ground, you can use a calculator or a computer program to solve the inequality $-16t^2 + 1,000 > 0$. Alternatively, you can use a graphing calculator to graph the function $f(t) = -16t^2 + 1,000$ and then find the values of $t$ for which the function is above the x-axis.

Q: What is the significance of the interval of time during which the object is above the ground?

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A: The interval of time during which the object is above the ground is significant because it tells us how long the object will be in the air before it hits the ground. This information can be useful in a variety of applications, such as calculating the time it takes for a parachute to deploy or determining the maximum height of a jump.

Q: Can I use this formula to calculate the distance of the object above the ground for any value of $t$?

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A: No, you cannot use this formula to calculate the distance of the object above the ground for any value of $t$. The formula is only valid for values of $t$ that satisfy the inequality $-16t^2 + 1,000 > 0$. For values of $t$ that do not satisfy this inequality, the formula will not give you the correct distance.

Q: How do I know if the object is above the ground or not?

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A: To determine if the object is above the ground or not, you can use the formula $d = -16t^2 + 1,000$ and calculate the distance of the object above the ground for a given value of $t$. If the distance is greater than zero, then the object is above the ground. If the distance is less than or equal to zero, then the object is not above the ground.

Q: Can I use this formula to calculate the distance of the object above the ground for multiple values of $t$?

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A: Yes, you can use this formula to calculate the distance of the object above the ground for multiple values of $t$. Simply plug in the different values of $t$ into the formula and calculate the corresponding distances.

Q: How do I graph the function $f(t) = -16t^2 + 1,000$?

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A: To graph the function $f(t) = -16t^2 + 1,000$, you can use a graphing calculator or a computer program. Simply enter the function into the calculator or program and then adjust the window settings to get a clear view of the graph.

Q: What is the x-intercept of the graph of the function $f(t) = -16t^2 + 1,000$?

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A: The x-intercept of the graph of the function $f(t) = -16t^2 + 1,000$ is the value of $t$ at which the function crosses the x-axis. This value of $t$ is given by $t = \sqrt{62.5}$.

Q: What is the y-intercept of the graph of the function $f(t) = -16t^2 + 1,000$?

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A: The y-intercept of the graph of the function $f(t) = -16t^2 + 1,000$ is the value of $f(t)$ at $t = 0$. This value is given by $f(0) = 1,000$.

Q: Can I use this formula to calculate the distance of the object above the ground for a value of $t$ that is not an integer?

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A: Yes, you can use this formula to calculate the distance of the object above the ground for a value of $t$ that is not an integer. Simply plug in the non-integer value of $t$ into the formula and calculate the corresponding distance.

Q: How do I determine if the object is above the ground or not for a value of $t$ that is not an integer?

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A: To determine if the object is above the ground or not for a value of $t$ that is not an integer, you can use the formula $d = -16t^2 + 1,000$ and calculate the distance of the object above the ground for the given value of $t$. If the distance is greater than zero, then the object is above the ground. If the distance is less than or equal to zero, then the object is not above the ground.

Q: Can I use this formula to calculate the distance of the object above the ground for a value of $t$ that is negative?

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A: Yes, you can use this formula to calculate the distance of the object above the ground for a value of $t$ that is negative. Simply plug in the negative value of $t$ into the formula and calculate the corresponding distance.

Q: How do I determine if the object is above the ground or not for a value of $t$ that is negative?

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A: To determine if the object is above the ground or not for a value of $t$ that is negative, you can use the formula $d = -16t^2 + 1,000$ and calculate the distance of the object above the ground for the given value of $t$. If the distance is greater than zero, then the object is above the ground. If the distance is less than or equal to zero, then the object is not above the ground.

Q: Can I use this formula to calculate the distance of the object above the ground for a value of $t$ that is a fraction?

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A: Yes, you can use this formula to calculate the distance of the object above the ground for a value of $t$ that is a fraction. Simply plug in the fraction value of $t$ into the formula and calculate the corresponding distance.

Q: How do I determine if the object is above the ground or not for a value of $t$ that is a fraction?

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A: To determine if the object is above the ground or not for a value of $t$ that is a fraction, you can use the formula $d = -16t^2 + 1,000$ and calculate the distance of the object above the ground for the given value of $t$. If the distance is greater than zero, then the object is above the ground. If the distance is less than or equal to zero, then the object is not above the ground.

Q: Can I use this formula to calculate the distance of the object above the ground for a value of $t$ that is a decimal?

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A: Yes, you can use this formula to calculate the distance of the object above the ground for a value of $t$ that is a decimal. Simply plug in the decimal value of $t$ into the formula and calculate the corresponding distance.

Q: How do I determine if the object is above the ground or not for a value of $t$ that is a decimal?

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A: To determine if the object is above the ground or not for a value of $t$ that is a decimal, you can use the formula $d = -16t^2 + 1,000$ and calculate the distance of the object above the ground for the given value of $t$. If the distance is greater than zero, then the object is above the ground. If the distance is less than or equal to zero, then the object is not above the ground.

Q: Can I use this formula to calculate the distance of the object above the ground for a value of $t$ that is a complex number?

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A: No, you cannot use this formula to calculate the distance of the object above the ground for a value of $t$ that is a complex number. The formula is only valid for real values of $t$.

Q: How do I determine if the object is above the ground or not for a value of $t$ that is a complex number?

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A: You cannot use this formula to determine if the object is above the ground or not for a value of $t$ that is a complex number. The formula is only valid for real values of $t$.

Q: Can I use this formula to calculate the