By Kardi Teknomo, PhD.

AHP

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Quadratic Equation: Optimization without Calculus

In this section, I will show an illustrative example on simple optimization problem (to maximize some objective) and its solution using quadratic function.

Example

Suppose a farmer has a large piece of land and he wants to make a rectangular fence for his animals but he has no money to buy more wood for the fence. Therefore, the total length of the fence is fixed to be Quadratic Equation: Optimization without Calculus meters. What should the width and length of the rectangle such that the area is maximized?

Solution:

Suppose the width is denoted by Quadratic Equation: Optimization without Calculus and the length is denoted by Quadratic Equation: Optimization without Calculus . The perimeter of the rectangle is Quadratic Equation: Optimization without Calculus and the area of rectangle is Quadratic Equation: Optimization without Calculus . From the perimeter equation, we can write Quadratic Equation: Optimization without Calculus and inputting this into the area of rectangle, we obtain

Quadratic Equation: Optimization without Calculus .

This is a quadratic function in the form of Quadratic Equation: Optimization without Calculus where Quadratic Equation: Optimization without Calculus , Quadratic Equation: Optimization without Calculus with parameters Quadratic Equation: Optimization without Calculus , Quadratic Equation: Optimization without Calculus and Quadratic Equation: Optimization without Calculus .

Quadratic Equation: Optimization without Calculus

We will use the characteristic of quadratic function to solve this optimization problem. Since the quadratic parameter Quadratic Equation: Optimization without Calculus is negative, we have sad parabola with maximum extreme point at coordinate Quadratic Equation: Optimization without Calculus where Quadratic Equation: Optimization without Calculus is the discriminant. Inputting the values of parameters into the extreme point coordinate, we have

Quadratic Equation: Optimization without Calculus

The area of rectangle is maximized at Quadratic Equation: Optimization without Calculus square meter.

The length of the rectangle is Quadratic Equation: Optimization without Calculus meter

The width is computed as Quadratic Equation: Optimization without Calculus meter

Thus, the area of the fenced land is maximized if the boundary is a square with side of 62.5 meter.

Note: Knowing that square shape yield largest area when the perimeter is bounded by the amount of wood he has ( Quadratic Equation: Optimization without Calculus meters), now the farmer asks if he makes the region as circle, will he get larger area or smaller area. The perimeter of circle is Quadratic Equation: Optimization without Calculus and area of a circle is Quadratic Equation: Optimization without Calculus . Inputting Quadratic Equation: Optimization without Calculus into the equation of area of circle yields Quadratic Equation: Optimization without Calculus square meter, which is actually larger than a square with the same perimeter.

Notice that we do not use any derivative or calculus to solve the optimization problems above.

Go to the next section to find out more resources on quadratic function, equation and formula

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These tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi. (2008) Quadratic Function Tutorial . http://people.revoledu.com/kardi/tutorial/quadratic/