A fence must be built to enclose a rectangular area of 5000 ftsquared. fencing material costs $ 3 per foot for the two sides facing north and south and ​$6 per foot for the other two sides. find the cost of the least expensive fence.

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 sideand [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 footCost 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: