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This module is part of a collection of modules that were developed to support laboratory activities in a Precalculus for PreEngineers (MATH 1508) at the University of Texas at El Paso. Contained in this module of applications of quadratic equations in various fields of engineering and science. These include the motion of an object under constant acceleration, quantitative management, and break-even analysis.

Quadratic equations

Introduction

Quadratic equations play an important role in the modeling of many physical situations. Finding the roots of quadratic equations is a necessary skill. Being able to interpret these roots is an important ability that is important in understanding physical problems. In this module, we will present a number of applications of quadratic equations in several fields of engineering.

Determining the roots of quadratic equations

A quadratic equation has the following form

ax 2 + bx + c = 0 size 12{ ital "ax" rSup { size 8{2} } + ital "bx"+c=0} {}

Because a quadratic equation involves a polynomial of order 2, it will have two roots. In general, a quadratic equation will either have two roots that are both real or have two roots that are both complex. For the present module, we will restrict our attention to quadratic equations that have two real roots.

There are three methods that are effective in solving for the roots of a quadratic equation. They are:

  • Solution by factoring
  • Solution by completing the square
  • Solution by the quadratic formula

The applications that follow will include examples of each of these three methods of solution.

Motion of an object under uniform acceleration

We will begin our study of quadratic equations by considering an application that you will likely encounter later in physics and mechanical engineering classes. Let us consider an object that is subject to a uniform acceleration. By uniform, we mean an acceleration that is constant. Such an object might be an automobile, an aircraft, a rocket, etc. The motion of an object subjected to uniform acceleration can be expressed mathematically by the following equation.

s ( t ) = 1 2 a t 2 + v 0 t + s 0 size 12{s \( t \) = { {1} over {2} } `a`t rSup { size 8{2} } +v rSub { size 8{0} } t+s rSub { size 8{0} } } {}

where s ( t ) represents the position of the object as function of time t ,

a represents the constant acceleration of the object,

v 0 represents the value of the object’s velocity at time t = 0, and

s 0 represents the position of the object at time t = 0.

An equation of this sort is called an equation of motion . We will illustrate its use in the following exercise.

Example 1: For our first example, let us consider a dragster on a drag strip of length one-quarter mile. For time t<0, the dragster is at rest at the starting line. At time = 0, the driver depresses his gas pedal to produce a uniform acceleration of 50 m/s 2 . Under these conditions, how far will the dragster travel in 1 second?

Because the dragster travels in a horizontal direction, we will represent its distance from the starting point as a fuction of time as x ( t ). We also know that the value for the acceleration ( a ) is 30 m / s 2 . We can incorporate these changes in equation (1) to produce a new equation of motion for the dragster.

Questions & Answers

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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
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