Finding the Maximum Value of r

Oct 13, 2025 by thesweetdesigns.com 32 views

Understanding the Equation r = 5 + 10cos(θ)

The given equation is a polar equation, where r represents the distance from the origin to a point on a graph, and θ represents the angle between the positive x-axis and the line connecting the origin to that point. In this equation, r is a function of θ, and we are asked to find the zeros of r, the maximum value of r, and the minimum value of r.

Finding the Zeros of r

To find the zeros of r, we need to find the values of θ for which r = 0. In other words, we need to solve the equation 5 + 10cos(θ) = 0. This equation can be rewritten as 10cos(θ) = -5.

import numpy as np
def equation(theta):
return 10 * np.cos(theta) + 5
theta = np.arccos(-5/10)

However, there are two possible solutions to this equation: θ = π + arccos(-5/10) and θ = 2π - arccos(-5/10). This is because the cosine function has a period of 2π, and the arccosine function returns an angle between 0 and π.

# Define the two possible solutions
theta1 = np.pi + np.arccos(-5/10)
theta2 = 2 * np.pi - np.arccos(-5/10)

Therefore, there are two correct answers to the equation 5 + 10cos(θ) = 0.

Why are there 2 correct answers when arccos can only be 1?

The reason there are two correct answers is that the arccosine function returns an angle between 0 and π, but the cosine function has a period of 2π. This means that if we add or subtract 2π from the angle returned by the arccosine function, we get another angle that satisfies the equation.

For example, if we let θ = arccos(-5/10), then we know that cos(θ) = -5/10. But we also know that cos(θ + 2π) = cos(θ) = -5/10, and cos(θ - 2π) = cos(θ) = -5/10. Therefore, we can write θ = π + arccos(-5/10) and θ = 2π - arccos(-5/10) as two possible solutions to the equation.

Finding the Maximum Value of r

To find the maximum value of r, we need to find the maximum value of 5 + 10cos(θ). This occurs when cos(θ) = 1, which means that θ = 0.

# Define the maximum value of r
max_r = 5 + 10 * np.cos(0)

Therefore, the maximum value of r is 15.

Finding the Minimum Value of r

To find the minimum value of r, we need to find the minimum value of 5 + 10cos(θ). This occurs when cos(θ) = -1, which means that θ = π.

# Define the minimum value of r
min_r = 5 + 10 * np.cos(np.pi)

Therefore, the minimum value of r is -5.

Conclusion

In this article, we have discussed the equation r = 5 + 10cos(θ) and found the zeros of r, the maximum value of r, and the minimum value of r. We have shown that there are two correct answers to the equation 5 + 10cos(θ) = 0, and that the maximum value of r is 15 and the minimum value of r is -5.

Polar Equations and Graphs

Polar equations are a way of describing curves in the plane using the distance from the origin and the angle from the positive x-axis. They are often used to describe the shape of a curve, and can be used to create graphs of the curve.

In this article, we have used the polar equation r = 5 + 10cos(θ) to describe a curve in the plane. We have found the zeros of r, the maximum value of r, and the minimum value of r, and have shown that there are two correct answers to the equation 5 + 10cos(θ) = 0.

Applications of Polar Equations

Polar equations have many applications in mathematics and science. They are used to describe the shape of curves in the plane, and can be used to create graphs of the curve.

In physics, polar equations are used to describe the motion of objects in the plane. For example, the equation of a circle can be written in polar form as r = a, where a is the radius of the circle.

In engineering, polar equations are used to describe the shape of curves in the plane. For example, the equation of a parabola can be written in polar form as r = a(1 + cos(θ)), where a is a constant.

Conclusion

Polar equations are a powerful tool for describing curves in the plane, and have many applications in mathematics and science. They are used to describe the shape of curves in the plane, and can be used to create graphs of the curve.

References

"Polar Equations and Graphs" by Michael Corral
"Polar Coordinates" by Math Open Reference
"Polar Equations" by Wolfram MathWorld
Q&A: Polar Equations and Graphs

In this article, we will answer some common questions about polar equations and graphs.

Q: What is a polar equation?

A: A polar equation is an equation that describes a curve in the plane using the distance from the origin and the angle from the positive x-axis.

Q: How do I write a polar equation?

A: To write a polar equation, you need to use the variables r and θ, where r is the distance from the origin and θ is the angle from the positive x-axis. For example, the equation r = 2sin(θ) is a polar equation.

Q: What is the difference between a polar equation and a Cartesian equation?

A: A Cartesian equation is an equation that describes a curve in the plane using the variables x and y. A polar equation, on the other hand, uses the variables r and θ to describe a curve in the plane.

Q: How do I graph a polar equation?

A: To graph a polar equation, you need to use a graphing calculator or a computer program that can graph polar equations. You can also use a table of values to plot points on the graph.

Q: What are some common polar equations?

A: Some common polar equations include:

r = a (a circle)
r = a(1 + cos(θ)) (a cardioid)
r = a(1 + sin(θ)) (a limacon)
r = a(1 - cos(θ)) (a lemniscate)
r = a(1 - sin(θ)) (a rose curve)

Q: How do I find the zeros of a polar equation?

A: To find the zeros of a polar equation, you need to set the equation equal to zero and solve for θ. For example, to find the zeros of the equation r = 2sin(θ), you would set 2sin(θ) = 0 and solve for θ.

Q: How do I find the maximum and minimum values of a polar equation?

A: To find the maximum and minimum values of a polar equation, you need to find the maximum and minimum values of the function r = f(θ). For example, to find the maximum and minimum values of the equation r = 2sin(θ), you would find the maximum and minimum values of the function 2sin(θ).

Q: What are some applications of polar equations?

A: Polar equations have many applications in mathematics and science. Some examples include:

Describing the shape of curves in the plane
Modeling the motion of objects in the plane
Creating graphs of functions
Solving problems in physics and engineering

Q: How do I use polar equations in real-world applications?

A: To use polar equations in real-world applications, you need to understand how to write and graph polar equations, and how to use them to model real-world problems. For example, you could use polar equations to model the motion of a satellite in orbit around the Earth, or to describe the shape of a curve in a engineering design.

Q: What are some common mistakes to avoid when working with polar equations?

A: Some common mistakes to avoid when working with polar equations include:

Confusing the variables r and θ
Not using the correct units for the variables r and θ
Not checking the domain and range of the function
Not using the correct graphing techniques

Q: How do I troubleshoot common problems with polar equations?

A: To troubleshoot common problems with polar equations, you need to understand the common mistakes to avoid and how to use graphing techniques to visualize the graph. You can also use a graphing calculator or a computer program to help you visualize the graph and identify any errors.

Conclusion

In this article, we have answered some common questions about polar equations and graphs. We have discussed how to write and graph polar equations, how to find the zeros and maximum and minimum values of polar equations, and how to use polar equations in real-world applications. We have also discussed some common mistakes to avoid and how to troubleshoot common problems with polar equations.