Real World Applications of Cubic Equations

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SUMMARY

This discussion centers on the real-world applications of cubic and higher-order polynomial equations, specifically cubic (third degree) and septic (seventh degree) equations. Participants highlight that cubic equations are commonly used in modeling physical phenomena such as fluid dynamics, magnetic hysteresis, and mechanical systems like rubber-band airplanes. Additionally, cubic equations are relevant in business contexts, particularly in analyzing the relationship between asset acquisition costs and return on investment. The conversation also touches on the use of septic equations in CAD/CAM for creating smooth surfaces.

PREREQUISITES
  • Understanding of polynomial equations, specifically cubic and septic equations.
  • Familiarity with Taylor's theorem and its applications in approximating functions.
  • Basic knowledge of fluid dynamics and mechanical systems.
  • Awareness of CAD/CAM principles and their relevance in design.
NEXT STEPS
  • Research the applications of cubic equations in fluid dynamics and magnetic hysteresis.
  • Explore Taylor's theorem and its implications in function approximation.
  • Investigate the use of septic equations in CAD/CAM for surface modeling.
  • Examine case studies on the relationship between asset acquisition costs and ROI using polynomial functions.
USEFUL FOR

This discussion is beneficial for mathematicians, engineers, business analysts, and anyone interested in the practical applications of polynomial equations in various fields such as physics, engineering design, and finance.

longbusy
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I've been kicking myself trying to think of a few real world applications of cubic equations (and x^4 quintive?). Can anyone give me a few examples?

Thanks,
Jeremy
 
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Any continuous function (graph that you can draw without lifting your pen) can be approximated to arbitrary (as good as you like) precision by series of x, x^2, x^3, etc. This result is called Taylors theorem.

So think of any two quantitative things that are related, and draw a graph of their relationship. This graph can be approximated by (worked out with pencil and paper) a cubic or quintic equation (the higher the power of x the more accurate the approximation).
 
Cubic polynomials are often used in conjunction with splines.
Roots of (cubic) polynomials, i.e solutions of a cubic equation, are not, as such, "applicable".
 
longbusy said:
I've been kicking myself trying to think of a few real world applications of cubic equations (and x^4 quintive?). Can anyone give me a few examples?

Thanks,
Jeremy

Presumably you have examples of what you consider to be real world uses of quadratics and linear polys, then?

(fourth power is quartic, 5th quintic)
 
Actually, I do not have examples, that is my purpose here. I am sure fluid/air flow or other physical examples can be modeled according to these polynomial equations.
 
Your stated purpose was to know about cubics and higher. The implication being that you considered quadratics in the situations you have met, and only quadratics, or simpler.

If we knew what those "real life" equations were we might be able to explain things better here.

However, it should be clear to you that it is not unreasonable that anything having some volume dependence may well incoporate cubics. Or if the acceleration of a body varies linearly in time then equations of motion in terms of displacement wrt to time will involve possibly complicated cubic equations. The reciprocity law of resistance (I think I may have made that up, sorry, but the R_1R_2/2(R_1+R_2) isn't even a polynomial in R_1 or R_2. The equation of motion of a pendulum depends on solving differential equations that can't be done by elementary means, and hence we linearize them so that we may solve them.
 
Here's an interesting application of a cubic: put a bar of soft iron in a mild magnetic field. A slight magnetism is induced in the iron. As you increase the strength of the magnetic field slowly, the magnetism of the iron will increase slowly, but then suddenly jump up after which, as you still increase the strength of the magnetic field, it increases slowly again. If you now DECREASE the strength of the magnetic field, the magnetization will, of course, decrease slowly- PAST the strength at which is had suddenly jumped up. It will then suddenly jump down but at a point where the magnetic field is weaker than when it jumped up. E.e.s call that a "hysteresis loop".

Similar application. You wind the propellor of a toy rubber-band airplane and the rubber-band winds around itself. Suddenly, the entire rubber-band will twist into a spiral. If you now reverse the winding, the force on the rubber-band will go down PAST the point at which it had suddenly kinked before it "unkinks". That is referred to as the "rubber-band catastrophe" (anyone remember "catastrophe theory"?)

Why? Because the equilibrium solutions for magnetic field as a function of induced magnetization and for the force on the propellor as a function of "twist" of the rubber-band is a cubic. Notice the way those functions are going! Induced magnetization is not a FUNCTION of magnetic field (nor is "twist" a function of force) because the cubic would be "lying on its side" and we would have 3 values of induced magnetization for some values of magnetic field. Think of it as x= y3- 6y2+ 9y. The "switchback" section is between the two extrema for x, 4 and 18. In that region, the "switchback" section that connects the other two is an unstable equilibrium while the other two are stable. As you start increasing the magnetic field, you stay on the lower branch until you are past the local maximum x (in the example above, x= 18) and now the value jumps to the other branch. Reducing the magnetic field, you stay on the "upper" stable branch until you hit the local minimum x (in the example above, x= 4).
 
please help me.. anyone.. can u please tell me what is the real world application of septic equation? which the polynomial function with the degree of 7.. thank you~~~
 
Simplest answer would be volumes and space, like rectangular container measurements. You see these kinds of exercises (applications) in College Algebra, maybe even Intermediate Algebra.

Hey, k-style, for 7th degree polynomials, maybe that degree would best fit some sets of measured data; just depending on what the person analyzing the set finds to work best. No specific application comes to mind - just any set of points, (x, y) could be fit to some findable polynomial function.
 
  • #10
symbolipoint said:
Simplest answer would be volumes and space, like rectangular container measurements. You see these kinds of exercises (applications) in College Algebra, maybe even Intermediate Algebra.

Hey, k-style, for 7th degree polynomials, maybe that degree would best fit some sets of measured data; just depending on what the person analyzing the set finds to work best. No specific application comes to mind - just any set of points, (x, y) could be fit to some findable polynomial function.

thank you for answering.. i thought there would be any special situation that only septic/heptic equation can solve.. thanks anyway...
 
  • #11
I know that this is not a physics application but from the world of business I can offer an example of the practical application of a cubic equation.
In the rental business, it can be shown that the increase or decrease in the acquisition cost of an asset held for rental is related to the Return on Investment produced by the rental asset by a third order polynomial function.
 
  • #12
k-style said:
please help me.. anyone.. can u please tell me what is the real world application of septic equation? which the polynomial function with the degree of 7.. thank you~~~

I'm sure there are times in CAD/CAM when it would be advantageous to have a very 'smooth' (in a particular technical sense) surface modeled by a septic spline.
 
  • #13
Any easy example is in minimizing the material used to make a can. See for example the article on beer cans at http://www.math4realworld.com" .
 
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