The cost of the least expensive fence is required.
The cost of the least expensive fence is $1200.
Let [tex]x[/tex] be the length of the north and south facing side
and [tex]y[/tex] be the length of the other two sides.
The area is
[tex]A=xy\\\Rightarrow 5000=xy\\\Rightarrow y=\dfrac{5000}{x}[/tex]
Cost of north and south facing side is $3 per foot
Cost of other sides is $6 per foot.
Cost of the fence is
[tex]C=3(2x)+6(2y)\\\Rightarrow C=6x+\dfrac{6\times 2\times 5000}{x}\\\Rightarrow C=6x+\dfrac{60000}{x}[/tex]
Differentiating with respect to [tex]x[/tex]
[tex]C'=6-\dfrac{60000}{x^2}[/tex]
Equating with zero
[tex]0=6-\dfrac{60000}{x^2}\\\Rightarrow x=\sqrt{\dfrac{60000}{6}}\\\Rightarrow x=100[/tex]
Double derivative of the function is
[tex]C''=\dfrac{120000}{x^3}\\\Rightarrow C''(100)=\dfrac{120000}{100^3}=0.12>0[/tex]
Since, it is greater than zero the cost will be minimum at [tex]x=100[/tex]
Finding [tex]y[/tex]
[tex]y=\dfrac{5000}{100}\\\Rightarrow y=50[/tex]
The cost is
[tex]C=3\times 2\times 100+6\times2\times 50=1200[/tex]
Learn more:
https://brainly.com/question/19091228
