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Mathematics, optimization exercise : maximize the area of a field with a given perimeter. You will have to find the dimensions of the field (width and length, in meters) that has a maximum area with a perimeter of 1200 m. How to do so ? Maximizing the quadratic equation Area, using first and second derivative.

Derivative : solution exercise 1.3 (Advanced math level)

1.3 Optimization: A farmer wants to set up a fence around his field

A farmer whishes to build a fence around his rectangular field and to divide it in two equal parts as well. The dividing fence must be parallel to one of the side of the field. You are asked to find the dimensions of the field of maximum area that you can fence and divide in two equal parts with a wire fence of 1200 m.

Area maximization for a field with a given perimeter

(1) Area = x . y (in m2)

Fence length = 2y + 2x + x = 1200 m

1200 = 2y + 3x

y = (1200 - 3x)/2

(2) y = 600 - 1,5x

To find the width x of the field that makes the area maximum, you must calculate the derivative of the function Area, makes it zero and solve for x: Area ' = (x . y) ' = 0.

Area = x . y = x . (600 - 1,5x) by substituting equation (2) for y

Area = 600x - 1,5x2

(Area) ' = (600x - 1,5x2) ' = 0

= 600 - 3x = 0

600 = 3x

x = 200 m

We have to check that the x-value x = 200 corresponds to an extremum that is a maximum and not a minimum because we are trying to maximize the area of the field. However, in our case the second derivative does not have to be computed because the function
Area = 600x - 1,5x2 is a quadratic equation (second degree equation), thus it is the equation of a parabola. Moreover the factor of x2 is negative (its value is -1,5), which means that the parabola is concave down. Hence we know that the extremum of the parabola is a maximum. Everything is all right then: x = 200 m is the width of a field with an area that is maximum. This field can be fenced and divided into two equal parts using a 1200 meters long wire. The length of the field will be: y = 600 - 1,5x = 600 - 1,5 . 200 = 300 m.

Area derivative (derivative of a superficie)

Derivative : Advanced math level exercises