A farmer wants to fence an area of 37.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. what should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence

The fence cost will be optimized when the total length of fence in one direction (2 lengths of fence) is equal to the total length of fence in the other direction (3 lengths of fence). In order for that to be the case, the aspect ratio of the field must be 3:2. The short side will be √(2/3*37.5*10^6 ft²) = 5000 ft.

The long side will be 7500 ft, the short sides and the center partition will be 5000 ft each.

_____
Suppose the side of the field perpendicular to the partition is length x. Then the total length of fencing is
.. length = 2x + 3*(37500000/x)
This has a minimum at x=7500 ft. The corresponding total length of fence is 30,000 ft.

AkshayG AkshayG · Answer 2 · 2019-10-10T09:48:47+02:00

Thelengthof the sides of the rectangular filed tominimize the cost of the fenceis [tex]\boxed{{\mathbf{7500}}\,{\mathbf{ft}}}[/tex].

Further explanation:

It is given that the area of the fence is [tex]37.5\,{\text{ million f}}{{\text{t}}^2}[/tex] and divide it in half with a fence parallel to one of the sides of the rectangle as shown in Figure 1.

Consider the length of fence is [tex]x[/tex] and width of the fenceis [tex]y[/tex] .

Let [tex]A[/tex] be the area of the fence.

Now, area is the multiplication of length and width that is [tex]A = xy[/tex].

Here, area is given as [tex]37.5\,{\text{ million f}}{{\text{t}}^2}[/tex].

Substitute [tex]37.5\,{\text{ million f}}{{\text{t}}^2}[/tex] for [tex]A[/tex] in equation [tex]A = xy[/tex] as follows:

[tex]\begin{aligned}37500000 = xy\\y= \frac{{37500000}}{x}{\text{}}\\\end{aligned}[/tex] ......(1)

From Figure 1, it is observed that there are [tex]2x[/tex] and [tex]3y[/tex] so the expression for perimeter of the fence is [tex]P = 2x + 3y[/tex].

Substitute [tex]\dfrac{{37500000}}{x}[/tex] for [tex]y[/tex] in equation [tex]P = 2x + 3y[/tex].

[tex]\begin{aligned}P&= 2x + 3\left({\frac{{37500000}}{x}}\right)\\&= 2x +\frac{{112500000}}{x}{\text{}}\\\end{aligned}[/tex] ......(2)

Derivate the equation (2) with respect to [tex]x[/tex] as follows:

[tex]P' = 2 - \dfrac{{112500000}}{{{x^2}}}{\text{}}[/tex] ......(3)

Substitute [tex]0[/tex] for [tex]P'[/tex] in equation (3) to obtain the value of [tex]x[/tex].

[tex]\begin{aligned}\frac{{112500000}}{{{x^2}}}&=2\\{x^2}&=\frac{{112500000}}{2}\\{x^2}&= 56250000\\x&=\pm\,7500\\\end{aligned}[/tex]

Therefore, the value of [tex]x[/tex] is [tex]\,7500\,{\text{ or }} - \,7500[/tex].

Derivate the equation (3) as follows.

[tex]P'' = \dfrac{{112500000}}{{{x^3}}}[/tex]

Now, if the value of [tex]x[/tex] is positive, then [tex]P''[/tex] is alsopositive.

So, for obtaining the minimum dimension, the value of [tex]x[/tex] must be apositive number.

Therefore, the value of [tex]x[/tex] is [tex]{\mathbf{7500}}[/tex].

Substitute [tex]7500[/tex] for [tex]x[/tex] in equation (1) to obtain the value of [tex]y[/tex].

[tex]\begin{aligned}y&=\frac{{37500000}}{{7500}}\\&=5000\\\end{aligned}[/tex]

Thus, thelength of the sides of the rectangular filed to minimize the cost of the fence is [tex]\boxed{{\mathbf{7500}}\,{\mathbf{ft}}}[/tex].

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Answer Details:

Grade: Junior High School

Subject: Mathematics

Chapter: Surface Area and Volumes

Keywords:Surface area, linear equation, system of linear equations in two variables, rectangular filed, minimize the cost of the fence