A Rocket Is Launched From Atop A 36-foot Cliff With An Initial Velocity Of 107 Feet Per Second. The Height Of The Rocket, T T T Seconds After Launch, Is Given By The Equation H = − 16 T 2 + 107 T + 36 H = -16t^2 + 107t + 36 H = − 16 T 2 + 107 T + 36 . Use The Quadratic Formula To

Introduction

When a rocket is launched from a cliff, its trajectory can be described by a quadratic equation, which represents the height of the rocket as a function of time. In this article, we will use the quadratic formula to find the maximum height of a rocket launched from a 36-foot cliff with an initial velocity of 107 feet per second. The quadratic equation that describes the height of the rocket is given by $h = -16t^2 + 107t + 36$, where $h$ is the height of the rocket in feet and $t$ is the time in seconds.

The Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The quadratic formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In our case, the quadratic equation is $h = -16t^2 + 107t + 36$, which can be rewritten as $-16t^2 + 107t + 36 = 0$. We can now use the quadratic formula to find the values of $t$ that satisfy this equation.

Applying the Quadratic Formula

To apply the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the quadratic equation. In this case, we have:

$a = -16$ $b = 107$ $c = 36$

Now, we can plug these values into the quadratic formula:

$$t = \frac{-107 \pm \sqrt{107^2 - 4(-16)(36)}}{2(-16)}$$

Simplifying the expression under the square root, we get:

$$t = \frac{-107 \pm \sqrt{11449 + 2304}}{-32}$$

$$t = \frac{-107 \pm \sqrt{13753}}{-32}$$

$$t = \frac{-107 \pm 117}{-32}$$

Solving for $t$

Now, we have two possible values for $t$:

$$t_1 = \frac{-107 + 117}{-32} = \frac{10}{-32} = -\frac{5}{16}$$

$$t_2 = \frac{-107 - 117}{-32} = \frac{-224}{-32} = 7$$

Finding the Maximum Height

The maximum height of the rocket occurs when the time $t$ is equal to the larger of the two values we found. In this case, the maximum height occurs when $t = 7$ seconds.

To find the maximum height, we can plug this value of $t$ into the quadratic equation:

$$h = -16(7)^2 + 107(7) + 36$$

$$h = -16(49) + 749 + 36$$

$$h = -784 + 749 + 36$$

$$h = 1$$

Conclusion

In this article, we used the quadratic formula to find the maximum height of a rocket launched from a 36-foot cliff with an initial velocity of 107 feet per second. The quadratic equation that describes the height of the rocket is given by $h = -16t^2 + 107t + 36$, and we found that the maximum height occurs when $t = 7$ seconds. The maximum height of the rocket is 1 foot.

The Importance of Quadratic Equations

Quadratic equations are used to describe the trajectory of objects under the influence of gravity, such as rockets, projectiles, and even the motion of planets. By using the quadratic formula, we can find the maximum height of an object, as well as the time it takes to reach that height. This is a fundamental concept in physics and engineering, and is used to design and optimize the performance of various systems.

Real-World Applications

Quadratic equations have many real-world applications, including:

• Projectile motion: Quadratic equations are used to describe the trajectory of projectiles, such as bullets, arrows, and even the motion of a thrown ball.
• Rocket science: Quadratic equations are used to design and optimize the performance of rockets, including the calculation of maximum height and range.
• Engineering: Quadratic equations are used to design and optimize the performance of various systems, including bridges, buildings, and even the motion of vehicles.
• Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity, including the calculation of maximum height and range.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics and physics, and are used to describe the trajectory of objects under the influence of gravity. By using the quadratic formula, we can find the maximum height of an object, as well as the time it takes to reach that height. This is a fundamental concept in physics and engineering, and is used to design and optimize the performance of various systems.

Introduction

In our previous article, we used the quadratic formula to find the maximum height of a rocket launched from a 36-foot cliff with an initial velocity of 107 feet per second. In this article, we will answer some of the most frequently asked questions about the trajectory of a rocket and the use of quadratic equations in physics and engineering.

Q: What is the significance of the quadratic formula in physics and engineering?

A: The quadratic formula is a fundamental concept in physics and engineering, and is used to describe the trajectory of objects under the influence of gravity. It is used to calculate the maximum height and range of projectiles, as well as the time it takes to reach those heights.

Q: How is the quadratic formula used in rocket science?

A: The quadratic formula is used in rocket science to design and optimize the performance of rockets. It is used to calculate the maximum height and range of a rocket, as well as the time it takes to reach those heights. This information is critical in designing the trajectory of a rocket and ensuring that it reaches its intended destination.

Q: What is the difference between the quadratic formula and the quadratic equation?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The quadratic equation, on the other hand, is a mathematical expression that describes the relationship between the variables in a quadratic equation.

Q: How is the quadratic formula used in projectile motion?

A: The quadratic formula is used in projectile motion to describe the trajectory of a projectile under the influence of gravity. It is used to calculate the maximum height and range of a projectile, as well as the time it takes to reach those heights.

Q: What is the significance of the quadratic formula in engineering?

A: The quadratic formula is a fundamental concept in engineering, and is used to design and optimize the performance of various systems. It is used to calculate the maximum height and range of projectiles, as well as the time it takes to reach those heights.

Q: How is the quadratic formula used in physics?

A: The quadratic formula is used in physics to describe the motion of objects under the influence of gravity. It is used to calculate the maximum height and range of projectiles, as well as the time it takes to reach those heights.

Q: What is the difference between the quadratic formula and the quadratic equation in physics and engineering?

A: In physics and engineering, the quadratic formula and the quadratic equation are used to describe the same relationship between the variables in a quadratic equation. However, the quadratic formula provides the solutions to the quadratic equation, while the quadratic equation describes the relationship between the variables.

Q: How is the quadratic formula used in real-world applications?

A: The quadratic formula is used in a variety of real-world applications, including:

• Projectile motion: The quadratic formula is used to describe the trajectory of projectiles, such as bullets, arrows, and even the motion of a thrown ball.
• Rocket science: The quadratic formula is used to design and optimize the performance of rockets, including the calculation of maximum height and range.
• Engineering: The quadratic formula is used to design and optimize the performance of various systems, including bridges, buildings, and even the motion of vehicles.
• Physics: The quadratic formula is used to describe the motion of objects under the influence of gravity, including the calculation of maximum height and range.

Conclusion

In conclusion, the quadratic formula is a fundamental concept in physics and engineering, and is used to describe the trajectory of objects under the influence of gravity. It is used to calculate the maximum height and range of projectiles, as well as the time it takes to reach those heights. This information is critical in designing the trajectory of a rocket and ensuring that it reaches its intended destination.