An Object Is Launched From The Ground. The Object's Height, In Feet, Can Be Described By The Quadratic Function $h(t) = 80t - 16t^2$, Where $t$ Is The Time, In Seconds, Since The Object Was Launched. When Will The Object Hit The

Introduction

When an object is launched from the ground, its height can be described by a quadratic function. In this case, the height of the object, in feet, is given by the function $h(t) = 80t - 16t^2$, where $t$ is the time, in seconds, since the object was launched. The quadratic function is a polynomial of degree two, and it can be used to model the height of the object as a function of time. In this article, we will explore the properties of the quadratic function and determine when the object will hit the ground.

Understanding the Quadratic Function

The quadratic function $h(t) = 80t - 16t^2$ is a polynomial of degree two, which means that it has a leading term with a degree of two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, the quadratic function has a leading coefficient of $-16$, a linear coefficient of $80$, and a constant term of $0$. The quadratic function can be graphed as a parabola, which is a U-shaped curve that opens downward.

Graphing the Quadratic Function

To graph the quadratic function, we can use the fact that the vertex of the parabola is the point where the function has a maximum or minimum value. The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$. In this case, the vertex of the parabola is $x = -\frac{80}{2(-16)} = 2.5$. The y-coordinate of the vertex can be found by plugging this value into the function: $h(2.5) = 80(2.5) - 16(2.5)^2 = 200 - 100 = 100$. Therefore, the vertex of the parabola is the point $(2.5, 100)$.

Finding the Time When the Object Hits the Ground

To find the time when the object hits the ground, we need to find the value of $t$ when $h(t) = 0$. This means that we need to solve the equation $80t - 16t^2 = 0$. We can factor out a $t$ from the equation: $t(80 - 16t) = 0$. This gives us two possible solutions: $t = 0$ and $80 - 16t = 0$. The first solution, $t = 0$, corresponds to the time when the object was launched, which is not the time when the object hits the ground. The second solution, $80 - 16t = 0$, can be solved for $t$ by dividing both sides by $-16$: $t = \frac{80}{16} = 5$. Therefore, the object will hit the ground after $5$ seconds.

Conclusion

In this article, we explored the properties of the quadratic function $h(t) = 80t - 16t^2$ and determined when the object will hit the ground. We graphed the quadratic function as a parabola and found the vertex of the parabola, which is the point $(2.5, 100)$. We then solved the equation $80t - 16t^2 = 0$ to find the time when the object hits the ground, which is after $5$ seconds.

Additional Information

• The quadratic function $h(t) = 80t - 16t^2$ can be used to model the height of the object as a function of time.
• The vertex of the parabola is the point $(2.5, 100)$.
• The object will hit the ground after $5$ seconds.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath
• [3] "Solving Quadratic Equations" by Mathway

Discussion

• What is the significance of the vertex of the parabola in this problem?
• How does the quadratic function $h(t) = 80t - 16t^2$ relate to the height of the object?
• What is the physical significance of the time when the object hits the ground?

Related Topics

• Quadratic Functions
• Graphing Quadratic Functions
• Solving Quadratic Equations
• Physics of Projectile Motion
  

Introduction

In our previous article, we explored the properties of the quadratic function $h(t) = 80t - 16t^2$ and determined when the object will hit the ground. In this article, we will answer some of the most frequently asked questions about the quadratic function and the object's motion.

Q&A

Q: What is the significance of the vertex of the parabola in this problem?

A: The vertex of the parabola is the point where the function has a maximum or minimum value. In this case, the vertex of the parabola is the point $(2.5, 100)$, which represents the maximum height of the object.

Q: How does the quadratic function $h(t) = 80t - 16t^2$ relate to the height of the object?

A: The quadratic function $h(t) = 80t - 16t^2$ is a mathematical model that describes the height of the object as a function of time. The function takes into account the initial velocity of the object and the acceleration due to gravity.

Q: What is the physical significance of the time when the object hits the ground?

A: The time when the object hits the ground is the point at which the object's height is zero. This is the moment when the object's velocity is zero, and it is no longer in motion.

Q: Can you explain the concept of quadratic functions in simpler terms?

A: A quadratic function is a mathematical model that describes a relationship between two variables. In this case, the quadratic function $h(t) = 80t - 16t^2$ describes the relationship between the height of the object and the time since it was launched.

Q: How do you graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: Can you solve a quadratic equation using a calculator?

A: Yes, you can solve a quadratic equation using a calculator. Most calculators have a built-in function for solving quadratic equations.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial of degree two, while a linear function is a polynomial of degree one. A quadratic function has a parabolic shape, while a linear function has a straight line shape.

Q: Can you explain the concept of vertex in simpler terms?

A: The vertex of a parabola is the point where the function has a maximum or minimum value. It is the highest or lowest point on the graph.

Q: How do you find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. This formula gives you the x-coordinate of the vertex.

Q: Can you explain the concept of quadratic equations in simpler terms?

A: A quadratic equation is a mathematical equation that involves a quadratic function. It is an equation that can be written in the form $ax^2 + bx + c = 0$.

Q: How do you solve a quadratic equation?

A: To solve a quadratic equation, you can use factoring, the quadratic formula, or a calculator.

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

A: The coefficient of the quadratic term in a quadratic function determines the shape of the parabola. A positive coefficient results in a parabola that opens upward, while a negative coefficient results in a parabola that opens downward.

Q: Can you explain the concept of quadratic functions in the context of physics?

A: Quadratic functions are used in physics to model the motion of objects under the influence of gravity. The quadratic function $h(t) = 80t - 16t^2$ is an example of a quadratic function that models the height of an object as a function of time.

Conclusion

In this article, we answered some of the most frequently asked questions about the quadratic function and the object's motion. We hope that this article has provided a better understanding of the concepts involved and has helped to clarify any doubts that you may have had.

Additional Information

• Quadratic functions are used in physics to model the motion of objects under the influence of gravity.
• The vertex of a parabola is the point where the function has a maximum or minimum value.
• The quadratic function $h(t) = 80t - 16t^2$ is an example of a quadratic function that models the height of an object as a function of time.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath
• [3] "Solving Quadratic Equations" by Mathway

Discussion

• What is the significance of the coefficient of the quadratic term in a quadratic function?
• How do you graph a quadratic function?
• What is the difference between a quadratic function and a linear function?