Solving Logarithmic Spirals for Length from Origin to Point X

In summary, the conversation is about finding the length of a logarithmic spiral from the origin to a point X on the curve. The distance between the origin and point X is given, as well as the distance between the origin and point A, which is the same as the distance between points A, B, and C. The location of point X and the angle between the origin and point X are unknown. The equations used for solving this problem are r=ae^(bθ) and L=rθ. It is suggested to solve for theta first, then use L=rθ to find the total length. The values at points A and B do not directly correspond to the terms a and b in the equation.
  • #1
eXmag
36
0

Homework Statement



This is a logarithmic spiral. How can I find the length of the spiral (curve) from the origin to point X on the curve? The distance between the origin and the point is given (we can call that the radius) and the distance between the origin and point A is given which this is the same between points A and B and C. Point X is on the spiral with an indicated arrow pointing towards it. The location, therefore, the angle between the origin and this point is unknown. However the distance is known. How can I find the total length of the spiral from the origin to point X?

Mod note: The text above is from the oversized image originally posted. A cropped version of the image is below.

attachment.php?attachmentid=57047&stc=1&d=1364122742.jpg


Homework Equations



1. ##r=ae^{b\theta}##
2. ##L=r\theta##


The Attempt at a Solution



Do I need to solve for theta then use L=r(theta) to find the total length?
I'm having troubles finding the terms for the first equation.
 

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  • #2
Do I need to solve for theta then use L=r(theta) to find the total length?
That is a good idea.
Im having troubles finding the terms for the first equation.
You know that A and B satisfy the equation for the spiral. This allows to determine a and b.
 
  • #3
Are you referring to the points A and B? So my values at point A and B correspond to the a and b terms in the equation?
 
  • #4
eXmag said:
Are you referring to the points A and B?
Sure
So my values at point A and B correspond to the a and b terms in the equation?
No. The names are a bit misleading in that respect.
 

FAQ: Solving Logarithmic Spirals for Length from Origin to Point X

What is a logarithmic spiral?

A logarithmic spiral is a spiral curve that grows in a self-similar way, meaning that it maintains its shape as it expands or contracts. It is also known as an equiangular spiral because its angle with the polar axis remains constant as it grows.

How do you solve for the length from the origin to a point on a logarithmic spiral?

To solve for the length from the origin to a point on a logarithmic spiral, you can use the formula L = a * ln(r), where L is the length, a is a constant, and r is the distance from the origin to the point on the spiral. This formula can be derived using calculus and the properties of logarithms.

What is the relationship between the length from the origin to a point on a logarithmic spiral and its polar angle?

The length from the origin to a point on a logarithmic spiral is directly proportional to its polar angle. This means that the farther the point is from the origin, the larger the polar angle will be, and therefore, the longer the length will be. This relationship is expressed in the formula L = a * ln(r), where a is a constant representing the polar angle.

Can you use logarithmic spirals to model real-life phenomena?

Yes, logarithmic spirals can be found in nature and can be used to model various real-life phenomena. They are commonly seen in the shapes of seashells, galaxies, and hurricanes. They can also be used in architecture and design to create aesthetically pleasing and efficient structures.

What are some other applications of logarithmic spirals in mathematics and science?

Logarithmic spirals have various applications in mathematics and science. They are used in navigation and mapmaking to create accurate and efficient routes. They also have applications in signal processing, image compression, and antenna design. In physics, they are used to model electromagnetic fields and the behavior of fluids in turbulent flow. In biology, they can be used to study the growth patterns of plants and animals.

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