A Lighthouse Is Located At { (1,2)$}$ In A Coordinate System Measured In Miles. A Sailboat Starts At { (-7,8)$}$ And Sails In A Positive { X$}$-direction Along A Path That Can Be Modeled By A Quadratic Function With A

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Introduction

A lighthouse is a beacon of hope and safety for sailors navigating through treacherous waters. In this scenario, we have a lighthouse located at the coordinates (1,2) in a coordinate system measured in miles. A sailboat starts at the coordinates (-7,8) and sails in a positive x-direction along a path that can be modeled by a quadratic function. In this article, we will delve into the mathematical world of quadratic functions and explore how they can be used to model the sailboat's path.

Quadratic Functions: A Brief Overview

Quadratic functions are a type of polynomial function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. These functions have a parabolic shape and can be used to model a wide range of phenomena, from the trajectory of a projectile to the growth of a population.

In our scenario, the sailboat's path can be modeled by a quadratic function of the form f(x) = ax^2 + bx + c, where x represents the distance traveled in the positive x-direction. The constants a, b, and c can be determined using the coordinates of the sailboat and the lighthouse.

Determining the Constants

To determine the constants a, b, and c, we need to use the coordinates of the sailboat and the lighthouse. Let's assume that the sailboat starts at the coordinates (-7,8) and sails in a positive x-direction. The lighthouse is located at the coordinates (1,2).

We can use the coordinates of the sailboat and the lighthouse to write a system of equations. Let's say that the sailboat's path is modeled by the quadratic function f(x) = ax^2 + bx + c. We can plug in the coordinates of the sailboat and the lighthouse into the function to get two equations:

f(-7) = a(-7)^2 + b(-7) + c = 49a - 7b + c = 8 f(1) = a(1)^2 + b(1) + c = a + b + c = 2

We can solve this system of equations to determine the values of a, b, and c.

Solving the System of Equations

To solve the system of equations, we can use substitution or elimination. Let's use substitution. We can solve the second equation for c:

c = 2 - a - b

We can plug this expression for c into the first equation:

49a - 7b + (2 - a - b) = 8

Simplifying the equation, we get:

48a - 8b = 6

We can solve this equation for a:

a = (6 + 8b) / 48

We can plug this expression for a into the second equation:

(6 + 8b) / 48 + b + (2 - (6 + 8b) / 48 - b) = 2

Simplifying the equation, we get:

(6 + 8b) / 48 + (2 - (6 + 8b) / 48) = 2

We can solve this equation for b:

b = 1

We can plug this value of b into the expression for a:

a = (6 + 8(1)) / 48

Simplifying the equation, we get:

a = 19/48

We can plug this value of a into the expression for c:

c = 2 - (19/48) - 1

Simplifying the equation, we get:

c = 25/48

The Sailboat's Path

Now that we have determined the values of a, b, and c, we can write the quadratic function that models the sailboat's path:

f(x) = (19/48)x^2 + x + (25/48)

This function represents the sailboat's path as it sails in a positive x-direction.

The Lighthouse's Location

The lighthouse is located at the coordinates (1,2). We can use the quadratic function to find the distance between the sailboat and the lighthouse at any given time.

Let's say that the sailboat is at the coordinates (x, f(x)). We can use the distance formula to find the distance between the sailboat and the lighthouse:

d = sqrt((x - 1)^2 + (f(x) - 2)^2)

We can plug in the expression for f(x) to get:

d = sqrt((x - 1)^2 + ((19/48)x^2 + x + (25/48) - 2)^2)

Simplifying the equation, we get:

d = sqrt((x - 1)^2 + ((19/48)x^2 + x - 31/48)^2)

Conclusion

In this article, we have explored the mathematical world of quadratic functions and used them to model the sailboat's path. We have determined the values of a, b, and c using the coordinates of the sailboat and the lighthouse. We have also used the quadratic function to find the distance between the sailboat and the lighthouse at any given time.

Quadratic functions are a powerful tool for modeling a wide range of phenomena. They can be used to model the trajectory of a projectile, the growth of a population, and many other things. In this scenario, we have used a quadratic function to model the sailboat's path as it sails in a positive x-direction.

References

Further Reading

Introduction

In our previous article, we explored the mathematical world of quadratic functions and used them to model the sailboat's path. We determined the values of a, b, and c using the coordinates of the sailboat and the lighthouse. We also used the quadratic function to find the distance between the sailboat and the lighthouse at any given time.

In this article, we will answer some of the most frequently asked questions about the sailboat's path and the lighthouse's location.

Q&A

Q: What is the equation of the sailboat's path?

A: The equation of the sailboat's path is f(x) = (19/48)x^2 + x + (25/48).

Q: What are the coordinates of the sailboat and the lighthouse?

A: The sailboat starts at the coordinates (-7,8) and sails in a positive x-direction. The lighthouse is located at the coordinates (1,2).

Q: How do you find the distance between the sailboat and the lighthouse at any given time?

A: To find the distance between the sailboat and the lighthouse at any given time, you can use the distance formula:

d = sqrt((x - 1)^2 + (f(x) - 2)^2)

where x is the distance traveled by the sailboat in the positive x-direction.

Q: What is the significance of the quadratic function in this scenario?

A: The quadratic function is used to model the sailboat's path as it sails in a positive x-direction. It represents the relationship between the distance traveled by the sailboat and its position at any given time.

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

A: A quadratic function is a type of polynomial function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. It has a parabolic shape and can be used to model a wide range of phenomena, from the trajectory of a projectile to the growth of a population.

Q: How do you determine the values of a, b, and c in a quadratic function?

A: To determine the values of a, b, and c in a quadratic function, you can use the coordinates of the sailboat and the lighthouse. You can plug in the coordinates into the function to get a system of equations, which you can then solve to find the values of a, b, and c.

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

A: Quadratic functions have many real-world applications, including modeling the trajectory of a projectile, the growth of a population, and the motion of an object under the influence of gravity.

Conclusion

In this article, we have answered some of the most frequently asked questions about the sailboat's path and the lighthouse's location. We have also explored the concept of quadratic functions and their significance in this scenario.

Quadratic functions are a powerful tool for modeling a wide range of phenomena. They can be used to model the trajectory of a projectile, the growth of a population, and many other things. In this scenario, we have used a quadratic function to model the sailboat's path as it sails in a positive x-direction.

References

Further Reading