Answer to Question #108765 in Calculus for mya

Question #108765
A fence must be built to enclose a rectangular area of 45,000ft^2. Fencing material cost $1 per foot for the two sides facing north and south and $2 per foot for the other two sides. Find the cost of the least expensive fence.
1
Expert's answer
2020-04-10T17:15:38-0400

Let "x=" the width of rectangle in ft, "y=" the length of rectangle in ft.

A fence must be built to enclose a rectangular area of 45,000 ft2

"xy=45000=>y={45000 \\over x}"

The cost of the fence


"C=2(2x)+1(2y)=4x+2y"

Hence


"C=C(x)=4x+2({45000 \\over x}),\\ x>0"

Find the first derivative with respect to "x"


"C'(x)=4-{90000 \\over x^2}"

Find the critical value(s)


"C'(x)=0=>4-{90000 \\over x^2}=0=>x_1=-150, x_2=150"

First derivative test

If "x<-150," "C'(x)>0, C(x)" increases.

If "-150<x<0,C'(x)<0,C(x)" decreases.

If "0<x<150, C'(x)<0,C(x)" decreases.

If "x>150,C'(x)>0, C(x)" increases.

Since "x>0," then the function "C(x)" has the absolute minimum at "x=150."

So "y=\\dfrac{45000}{150}\t=300"

The cost of the least expensive fence will be


"C_{min}=2(2)(150)+1(2)(300)=\\$1200"


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