A rectangle has the width x and its length is 10 inches greater than the width. For what values of x are the area of the rectangle greater or equal to 24 square inches?
Its length is 10 inches greater than the width. Therefore;
For what values of x, the area of the rectangle must be greater or equal to 24?
Factors are -2 and 12
Therefore;
Either;
Or;
-12 is outside the realm of possibilities.
Therefore the values of x must be greater or equal to 2.
Dividing all the terms with 4 gives;
Formula for general solution to quadratic equation is;
From our equation;
The solution to the quadratic equation
is a complex number.
Thus, there is no real number solution.
Therefore;
Take all terms to one side and collect like terms
Using the quadratic equation formula;
The value of x is a composite number;
Equate the two side to obtain value for x.
Use quadratic formula;
Applications of Quadratic and Cubic Equations in the Real World
There are numerous application of both quadratic and cubic equations in the real world. Typical real world examples where quadratic equations are used is in the modelling the speed of a car or vehicle and in calculating the profit of a business. On the other hand, cubic equations are widely used find the resistance of semi-conductors. They are also used to describe the motion of planets through space. This section will discuss the real word applications of quadratic and cubic equation. The paper will also provide examples that supports the application.
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Quadratic equations can be used to model the stopping distance of a vehicle or a moving object. The stopping distance refers to the distance travelled from the time the brakes are applied (reaction time) to the point when the car has stopped. Once the brakes are applied, the vehicle starts to decelerate. In general, the model is represented below;
Where bx relates to the thinking distance, ax^2 component is non-linear. The figures for braking distances as well as distance travelled during reaction time can be used to model a general quadratic equation for calculating the overall stopping distance. Let’s assume that we have modelled our equation and looks as shown below;
Suppose we know the stopping distance, C, we can find the value of x which is the speed the car is travelling. Let’s assume C is 75. Let the model be;
Now let’s solve the equation and determine the value of x.
Multiplying all terms by 20 gives;
Factors are -30 and 50.
Either:
Or;
Thus, the speed of the car is 30 mph.
Quadratic equations can also be used in calculating profits of business (Wandrei, 2018). For instance, if you want to sell a product and maximize profits, a quadratic equation can be used to model the profit for your product. Let’s take our demand model to be;
Assuming that the cost per unit is $10 (no fixed costs), we can use the equation above to obtain the equation of profits.
Substituting;
Collecting like terms
This equation can only be solved graphically. Let assume that we have obtained the maximum price as $35. This means that you have to make 500 units. The profits made from selling this units is $12,500.
There are numerous applications of Cubic equations. However, in this section I will discuss how engineers use cubic equations to find the resistance of semi-conductors as well as how they are used to describe the motion of planets through space. The formula used to calculate the resistance of a semi-conductor is as shown below (Malcolm, 2015);
Using Keplar’s third law of planetary motion, the motion of planets through space can be described. The law states that “the square of the orbital period of a planet is proportional to the cube of the mean distance from the sun” (Stern, 2014). The equation is stated below;
References
Malcolm, M. (2015). Cubic Equations in the Real World. [Online]. Available at: https://www.youtube.com/watch?v=ZogDCOF1zC4 . Accessed 8 th Dec 2018.
Stern, D. (2014). Kepler’s Three Laws of Planetary Motion. [Online]. Available at: http://www.phy6.org/stargaze/Kep3laws.htm . Accessed 8 th Dec 2018.
Wandrei, K. (2018). Everyday Examples of Situations to Apply Quadratic Equations. [Online]. Available at: https://sciencing.com/everyday-examples-situations-apply-quadratic-equations-10200.html . Accessed 8 th Dec 2018.
