From a tract of land a developer plans to fence a rectangular lot and then divide the lot into 2 identical smaller rectangular lots by putting a fence down the middle of the original lot. Suppose that the fence for the outside boundary costs $5 per foot and the fence down the middle costs $2 per foot. If each small lot is to contain 13,500 square feet, calculate the absolute minimum cost for which the fence for this project can be constructed. The absolute minimum value for the cost of constructing a fence that meets the above specifications is __________ dollars ?

Question

23-10-2016
Mathematics

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From a tract of land a developer plans to fence a rectangular lot and then divide the lot into 2 identical smaller rectangular lots by putting a fence down the middle of the original lot. Suppose that the fence for the outside boundary costs $5 per foot and the fence down the middle costs $2 per foot. If each small lot is to contain 13,500 square feet, calculate the absolute minimum cost for which the fence for this project can be constructed. The absolute minimum value for the cost of constructing a fence that meets the above specifications is __________ dollars ?

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alsteva alsteva · Answer 1 · 2016-11-05T07:58:06+01:00

Area of the big rectangle:
[tex]xy=27000 \\y= \frac{27000}{x} [/tex]

Total length of the fence:
[tex]\\P=2(x+y)+x \\P=2x+2y+x \\P=3x+2y [/tex]

Create function [tex]P(x)[/tex]:
[tex]\\P(x)=3x+2 \times \frac{27000}{x} \\P(x)= 3x+ \frac{54000}{x} [/tex]

Find the minimum value for [tex]x[/tex]:
[tex]\\P'(x)=3- \frac{54000}{x^2} \\P'(x)=0 \\3- \frac{54000}{x^2}=0 \\3x^2-54000=0 \\x^2=18000 \\x=\sqrt{18000} [/tex]

Find the minimum value for [tex]y[/tex]:
[tex]\\y= \frac{27000}{x} =\frac{27000}{\sqrt{18000}}[/tex]

Calculate the absolute minimum cost for which the fence for this project can be constructed:
[tex]\\C=2(x+y) \times \$5 + x \times \$2 \\C=2(\sqrt{18000}+\frac{27000}{\sqrt{18000}})\times \$5+\sqrt{18000} \times \$2 \\C \approx \$3622} [/tex]