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Name: Denise H.
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I am looking for some real life applications of solving quadratic equations by finding square roots. Other than the typical falling object model.


There are a variety.

One that comes to mind is gravitational force at planetary and stellar distances. Newton's force formula at a long distance is an inverse-square law. If you know the masses of a planet and its moon, as well as the force between them, you can calculate the distance with inverting a fraction and then taking the square root: (Force)=G(mass 1)(mass 2)/(distance)^2, where G=6.67x10^-11 J.m^2/kg^2.

A similar problem deals with centripetal force and velocity. For any object moving along a circular path, the force provided by the outside world must be just large enough to equal (Net Force)=(mass)(speed)^2/(path radius). If you know the mass of the Moon, the radius of its orbit, and the pull the Moon feels from the Earth, you can use the centripetal force equation to solve for the speed of the Moon. This same relation applies to a rock on a string spinning in a circle. Tension in the string is the force.

Another relation with a squared term is almost any circular or square area problem. A good example is the force exerted on such a surface by a liquid or gas. The pressure of the fluid multiplied by the area of the surface is the force exerted on that surface. The pressure of the air is 14.5 pounds per square inch, or 10^5 Newtons per square meter. A circular surface feels 30 pounds of pressure from the air. What is the radius of this circle?

Kinetic energy, or energy of motion, is another option. The formula for kinetic energy is KE=(1/2)(mass)(speed)^2. Any distance-squared, speed-squared, or 1/(distance-squared) relation will do just fine.

Dr. Ken Mellendorf
Physics Instructor
llinois Central College

There are two questions here: 1. "solving quadratic equations by finding square roots" and 2. Applications "other than the typical falling object model." Regarding 1. There are many ways to solve quadratic equations (equations in which the highest power of the unknown or unknowns is 2) other than by directly taking square roots. The familiar "quadratic formula", "completing the square", so called "Newton's method", graphing, and a variety of numerical methods. Most quadratic equations are NOT in the form of a "perfect square" so that taking the square root directly provides the solution.

Regarding 2. Applications of quadratic equations are so numerous in all of science and mathematics that it is difficult to know where to start a list. Any problem involving circles, ellipses, and their 3 dimensional analogs the sphere and ellipsoid (of which there must be millions) all involve quadratic equations. The kinetic energy is defined as K.E.= 1/2 m*v^2 where 'v' is the velocity, so any problem involving kinetic energy (of which there must be billions) involve quadratic equations. Quadratics appear in all sorts of optical and acoustic phenomena (the parabolic mirror, or parabolic microphone), the rates of many chemical reactions vary as the square of the concentration of one or more of the reagents, a Gaussian distribution is of the form: ~ exp(-x^2) which is quadratic, so any application of statistics will involve quadratic equations in one form or another.

Vince Calder

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