Graph Vector Functions: Explaining Appearance of Graph on Sphere

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Homework Help Overview

The problem involves graphing a curve defined by parametric equations and explaining its appearance by demonstrating that it lies on a sphere. The subject area includes vector functions and geometric interpretation in three-dimensional space.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss manipulating the parametric equations to derive the equation of a sphere. There are questions about the relationship between the parametric equations and the geometric properties of a sphere.

Discussion Status

Some participants are actively exploring the implications of the parametric equations, with one participant attempting to substitute the variables into the sphere's equation to check for validity. There is an ongoing exchange of ideas without a clear consensus yet.

Contextual Notes

Participants are considering the relevance of the parameterization and the potential cancellation of terms in their calculations. There is a focus on understanding the geometric interpretation rather than just the algebraic manipulation.

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Homework Statement


Graph the curve with the parametric equations

x=√(1-0.25cos^2(10t)) * cost
y=√(1-0.25cos^2(10t)) * sint
z=0.5cos10t

explain the appearance of the graph by showing that it lies on a sphere


Homework Equations





The Attempt at a Solution


could anyone please just point me in the right direction on how to start this problem? any help would be greatly appreciated! thanks pf!
 
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'Graph the curve' seems pretty straightforward. Did you do that?

Did you manipulate the parametric equations for x, y, and z to see if you can obtain the equation of a sphere?

That would be pretty conclusive if you did.
 
A sphere is the set of points equidistant from the same point. If a sphere has radius R, what equation in x,y,z describes it? Take the x, y, z you were given, with the assorted sines and cosines and see what happens when you plug those variables into the equation of a sphere.

As far as graphing, I am a lousy artist, but if I had to graph it I'd sketch 3-d coordinates and graph some points based on cartesian x, y and z. This is maybe cheating, but I can't think too well in terms of cos 10t. The 10t will turn out to be irrelevant anyway-- try paragraph 1 and you'll see.
 
thank you brmath and steamking! brmath...that's exactly what I am trying now. I am plugging in x, y, and z into the equation of a sphere and seeing if my t's cancel and the equation is satisfied.
 

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