A Ball Is Thrown Into The Air From The Top Of A Building. Its Height, In Meters Above The Ground, As A Function Of Time In Seconds, Is Given By $h(t) = -4.9 T^2 + 22 T + 12$.How Long Does It Take To Reach Maximum Height? (Round Your Answer To

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Introduction

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When a ball is thrown into the air from the top of a building, its height above the ground can be modeled using a quadratic function. In this scenario, the height of the ball, in meters above the ground, is given by the function $h(t) = -4.9 t^2 + 22 t + 12$, where $t$ represents time in seconds. The goal is to determine how long it takes for the ball to reach its maximum height.

Understanding the Quadratic Function

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The given function $h(t) = -4.9 t^2 + 22 t + 12$ is a quadratic function, which can be written in the general form $h(t) = at^2 + bt + c$. In this case, the coefficients are $a = -4.9$, $b = 22$, and $c = 12$. The graph of a quadratic function is a parabola, which opens downward if $a$ is negative and upward if $a$ is positive.

Finding the Maximum Height

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To find the maximum height, we need to determine the vertex of the parabola. The vertex of a parabola is the point where the function reaches its maximum or minimum value. In this case, since the parabola opens downward, the vertex will represent the maximum height.

The x-coordinate of the vertex of a parabola can be found using the formula $x = -\frac{b}{2a}$. Plugging in the values of $a$ and $b$, we get:

$x = -\frac{22}{2(-4.9)}$

$x = -\frac{22}{-9.8}$

$x = 2.24$

This means that the ball reaches its maximum height at $t = 2.24$ seconds.

Calculating the Maximum Height

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To find the maximum height, we need to plug in the value of $t$ into the function $h(t) = -4.9 t^2 + 22 t + 12$. Substituting $t = 2.24$, we get:

$h(2.24) = -4.9 (2.24)^2 + 22 (2.24) + 12$

$h(2.24) = -4.9 (5.0176) + 49.28 + 12$

$h(2.24) = -24.67 + 49.28 + 12$

$h(2.24) = 36.61$

Therefore, the maximum height reached by the ball is approximately 36.61 meters.

Conclusion

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In conclusion, we have used the quadratic function $h(t) = -4.9 t^2 + 22 t + 12$ to model the height of a ball thrown into the air from the top of a building. By finding the vertex of the parabola, we determined that the ball reaches its maximum height at $t = 2.24$ seconds, with a maximum height of approximately 36.61 meters.

Discussion

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The calculation of the maximum height of a ball thrown into the air is an important problem in physics and engineering. It has numerous applications in fields such as projectile motion, ballistics, and aerospace engineering. The quadratic function provides a simple and effective way to model the height of the ball, allowing us to determine the maximum height and the time it takes to reach it.

References

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• [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
• [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Future Work

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In future work, we can explore other applications of quadratic functions in physics and engineering, such as modeling the motion of objects under the influence of gravity or friction. We can also investigate the use of quadratic functions in other fields, such as economics and finance.

Code

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The following Python code can be used to calculate the maximum height of the ball:

import numpy as np

# Define the function h(t) = -4.9 t^2 + 22 t + 12
def h(t):
    return -4.9 * t**2 + 22 * t + 12

# Define the time at which the ball reaches its maximum height
t_max = 2.24

# Calculate the maximum height
h_max = h(t_max)

print("The maximum height reached by the ball is approximately {:.2f} meters.".format(h_max))

This code defines the function $h(t) = -4.9 t^2 + 22 t + 12$ and calculates the maximum height by plugging in the value of $t = 2.24$. The result is printed to the console.

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Introduction

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In our previous article, we explored the problem of a ball thrown into the air from the top of a building, with its height above the ground modeled by the quadratic function $h(t) = -4.9 t^2 + 22 t + 12$. We determined that the ball reaches its maximum height at $t = 2.24$ seconds, with a maximum height of approximately 36.61 meters. In this article, we will answer some common questions related to this problem.

Q&A

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Q: What is the significance of the quadratic function in modeling the height of the ball?

A: The quadratic function provides a simple and effective way to model the height of the ball, allowing us to determine the maximum height and the time it takes to reach it. The quadratic function is a fundamental concept in physics and engineering, and it has numerous applications in fields such as projectile motion, ballistics, and aerospace engineering.

Q: How do we determine the maximum height of the ball?

A: To determine the maximum height of the ball, we need to find the vertex of the parabola represented by the quadratic function. The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$. Plugging in the values of $a$ and $b$, we get $x = 2.24$, which represents the time at which the ball reaches its maximum height.

Q: What is the maximum height reached by the ball?

A: The maximum height reached by the ball is approximately 36.61 meters.

Q: How long does it take for the ball to reach its maximum height?

A: The ball reaches its maximum height at $t = 2.24$ seconds.

Q: What is the significance of the coefficient $a$ in the quadratic function?

A: The coefficient $a$ represents the rate at which the height of the ball changes with respect to time. In this case, $a = -4.9$, which means that the height of the ball decreases at a rate of 4.9 meters per second squared.

Q: How do we calculate the maximum height of the ball using the quadratic function?

A: To calculate the maximum height of the ball, we need to plug in the value of $t$ into the function $h(t) = -4.9 t^2 + 22 t + 12$. Substituting $t = 2.24$, we get $h(2.24) = 36.61$ meters.

Q: What are some real-world applications of the quadratic function in physics and engineering?

A: The quadratic function has numerous applications in fields such as projectile motion, ballistics, and aerospace engineering. It is used to model the motion of objects under the influence of gravity or friction, and it is also used to design and optimize systems such as bridges, buildings, and aircraft.

Conclusion

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In conclusion, the quadratic function provides a simple and effective way to model the height of a ball thrown into the air. By determining the vertex of the parabola, we can find the maximum height and the time it takes to reach it. The quadratic function has numerous applications in physics and engineering, and it is a fundamental concept in understanding the motion of objects under the influence of gravity or friction.

Discussion

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The quadratic function is a fundamental concept in physics and engineering, and it has numerous applications in fields such as projectile motion, ballistics, and aerospace engineering. It is used to model the motion of objects under the influence of gravity or friction, and it is also used to design and optimize systems such as bridges, buildings, and aircraft.

References

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• [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
• [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Future Work

---

In future work, we can explore other applications of the quadratic function in physics and engineering, such as modeling the motion of objects under the influence of gravity or friction. We can also investigate the use of the quadratic function in other fields, such as economics and finance.

Code

---

The following Python code can be used to calculate the maximum height of the ball:

import numpy as np

# Define the function h(t) = -4.9 t^2 + 22 t + 12
def h(t):
    return -4.9 * t**2 + 22 * t + 12

# Define the time at which the ball reaches its maximum height
t_max = 2.24

# Calculate the maximum height
h_max = h(t_max)

print("The maximum height reached by the ball is approximately {:.2f} meters.".format(h_max))

This code defines the function $h(t) = -4.9 t^2 + 22 t + 12$ and calculates the maximum height by plugging in the value of $t = 2.24$. The result is printed to the console.