Finding the Length of a Path on a 3D Graph

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SUMMARY

The discussion focuses on calculating the length of a path on a 3D graph defined by the equation z = f(x, y) = cos(y) - cos(x). The user seeks to determine the length of a route from the origin to the point (4π, 0, 0) while maintaining a constant elevation. Additionally, the user is tasked with finding a minimal length route that follows the gradient starting in the positive x direction. The solution involves using integration techniques, specifically arc length formulas, to derive the necessary path lengths.

PREREQUISITES
  • Understanding of 3D graphing and functions
  • Familiarity with arc length formulas in calculus
  • Basic knowledge of gradients and their applications
  • Introduction to integration techniques
NEXT STEPS
  • Study the arc length formula for curves in 3D space
  • Learn about gradient vectors and their significance in optimization
  • Explore integration techniques relevant to path length calculations
  • Review differential equations to prepare for advanced calculus topics
USEFUL FOR

Students in calculus or multivariable calculus courses, particularly those studying 3D graphing, path optimization, and integration methods. This discussion is especially beneficial for anyone preparing for assessments involving these mathematical concepts.

david90
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z=f(x,y)= (cos y) - (cos x)

u are at the origin and want to hike to point (4pie,0,0). U want to get to 4pie,0,0 by hiking a route that alway keeps you at the same elevation. what is the length of this route?

I ploted the graph and I already know the route that will keep me at the same elevation from origin to 4pie,0,0 but how do I find the length of that route??

As for the second part

Your second route always moves along the gradient. Determine such a route of minimal lenght, assuming you start hiking in the positive x direction. What is the length? If you cannot find and exact answer, then determine an upper bound and a lower bound bbetween which the actual length must lie.

I have no clue on doing the second part.
I need some serious help because this will determine my final grade so be clear and help me as much as u can. Thanks
 
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it is an ec assigment that worth quite a lot, not a take home exam.

why do u have to use the arc formula? The path is not on the arc but on the path between the circle.

My teacher told me to do integration to get the answer. I haven't done differential equation yet so...

don't be shy with giving hints :wink:
 
Last edited:

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