A Dolphin Jumps Out Of The Water. The Quadratic Function Y = − 16 X 2 + 20 X Y = -16x^2 + 20x Y = − 16 X 2 + 20 X Models The Dolphin's Height Above The Water After X X X Seconds. How Long Is The Dolphin Out Of The Water? (Hint: The Height Of The Water Should Be 0.)A. 1.25

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous real-world applications. In this article, we will explore how to use quadratic equations to model a real-world scenario, specifically the height of a dolphin above the water after a certain time. We will use the given quadratic function $y = -16x^2 + 20x$ to find the time it takes for the dolphin to jump out of the water.

Understanding the Quadratic Function

The quadratic function $y = -16x^2 + 20x$ represents the height of the dolphin above the water after $x$ seconds. The coefficient of the $x^2$ term, $-16$, represents the rate at which the height decreases, while the coefficient of the $x$ term, $20$, represents the rate at which the height increases. The constant term, $0$, represents the initial height of the dolphin above the water.

Setting Up the Equation

To find the time it takes for the dolphin to jump out of the water, we need to set up an equation where the height of the water is equal to $0$. This means that we need to set $y = 0$ and solve for $x$.

Solving the Quadratic Equation

To solve the quadratic equation $-16x^2 + 20x = 0$, we can start by factoring out the greatest common factor, which is $4x$. This gives us:

$$-16x^2 + 20x = 4x(-4x + 5) = 0$$

Using the Zero Product Property

The zero product property states that if the product of two factors is equal to $0$, then at least one of the factors must be equal to $0$. In this case, we have:

$$4x(-4x + 5) = 0$$

This means that either $4x = 0$ or $-4x + 5 = 0$.

Solving for x

Solving for $x$ in the first equation, we get:

$$4x = 0 \Rightarrow x = 0$$

This represents the initial time when the dolphin is at the surface of the water.

Solving for $x$ in the second equation, we get:

$$-4x + 5 = 0 \Rightarrow -4x = -5 \Rightarrow x = \frac{5}{4} = 1.25$$

This represents the time it takes for the dolphin to jump out of the water.

Conclusion

In this article, we used the quadratic function $y = -16x^2 + 20x$ to model the height of a dolphin above the water after a certain time. We set up an equation where the height of the water is equal to $0$ and solved for $x$ using the zero product property. We found that the time it takes for the dolphin to jump out of the water is $1.25$ seconds.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand curves.

Final Thoughts

In conclusion, quadratic equations are a powerful tool for modeling real-world scenarios. By using the quadratic function $y = -16x^2 + 20x$, we were able to find the time it takes for a dolphin to jump out of the water. This example illustrates the importance of quadratic equations in mathematics and their numerous real-world applications.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Quadratic Functions" by Khan Academy
• [3] "Real-World Applications of Quadratic Equations" by Wolfram MathWorld

Additional Resources

• [1] "Quadratic Equations" by MIT OpenCourseWare
• [2] "Quadratic Functions" by University of California, Berkeley
• [3] "Real-World Applications of Quadratic Equations" by Coursera
  

Introduction

In our previous article, we explored how to use quadratic equations to model the height of a dolphin above the water after a certain time. We used the quadratic function $y = -16x^2 + 20x$ to find the time it takes for the dolphin to jump out of the water. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the significance of the quadratic function $y = -16x^2 + 20x$ in this problem?

A: The quadratic function $y = -16x^2 + 20x$ represents the height of the dolphin above the water after $x$ seconds. The coefficient of the $x^2$ term, $-16$, represents the rate at which the height decreases, while the coefficient of the $x$ term, $20$, represents the rate at which the height increases.

Q: Why do we need to set $y = 0$ to find the time it takes for the dolphin to jump out of the water?

A: We need to set $y = 0$ to find the time it takes for the dolphin to jump out of the water because the height of the water is equal to $0$ at that time.

Q: How do we solve the quadratic equation $-16x^2 + 20x = 0$?

A: We can solve the quadratic equation $-16x^2 + 20x = 0$ by factoring out the greatest common factor, which is $4x$. This gives us:

$$-16x^2 + 20x = 4x(-4x + 5) = 0$$

Q: What is the zero product property, and how do we use it to solve the quadratic equation?

A: The zero product property states that if the product of two factors is equal to $0$, then at least one of the factors must be equal to $0$. In this case, we have:

$$4x(-4x + 5) = 0$$

This means that either $4x = 0$ or $-4x + 5 = 0$.

Q: What is the time it takes for the dolphin to jump out of the water?

A: The time it takes for the dolphin to jump out of the water is $1.25$ seconds.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand curves.

Additional Questions and Answers

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one.

Q: How do we graph a quadratic equation?

A: We can graph a quadratic equation by plotting the points on the coordinate plane and drawing a smooth curve through them.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the quadratic function reaches its maximum or minimum value.

Conclusion

In this article, we answered some frequently asked questions related to the topic of using quadratic equations to model the height of a dolphin above the water after a certain time. We hope that this article has provided you with a better understanding of quadratic equations and their applications in real-world scenarios.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Quadratic Functions" by Khan Academy
• [3] "Real-World Applications of Quadratic Equations" by Wolfram MathWorld

Additional Resources

• [1] "Quadratic Equations" by MIT OpenCourseWare
• [2] "Quadratic Functions" by University of California, Berkeley
• [3] "Real-World Applications of Quadratic Equations" by Coursera