A fence is to be built to enclose a rectangular area of 800 square feet. The fence along three sides is to be made of material that costs $5 per foot. The material for the fourth side costs $15 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built.

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rejkjavik rejkjavik · Answer 1 · 2019-11-08T05:45:40+01:00

Answer:

dc/dx = 0 when x = 20. hence fence is 20×40

Step-by-step explanation:

Given data:

Rectangular Area = 800 ft^2

three side material cost = $5 per ft

fourth side material cost is $15 per ft

let expensive side is x and other dimension assume to be y

Then cost is c

c = 5(x+ 2y) + 15 x

xy = 800, so y = 800/x

c =5(x+ 1600/x^2) + 15x

for minimum when dc/dx = 0

[tex]\frac{dc}{dx} = 5(1-\frac{1600}{x^2}) + 15 = 0[/tex]

[tex]5(1-\frac{1600}{x^2}) + 15 = 0[/tex]

[tex](1-\frac{1600}{x^2}) + 3 = 0[/tex]

[tex](1-\frac{1600}{x^2}) = - 3[/tex]

[tex]x^2 - 1600 = -3x^2[/tex]

[tex]4x^2 - 1600 = 0[/tex]

(2x-40)(2x+40) = 0

x = 20

dc/dx = 0 when x = 20. hence fence is 20×40