you are fencing a rectangular enclosure that will hold chickens and goats. the e

Question

you are fencing a rectangular enclosure that will hold chickens and goats. the enclosure will be split into two sections with a fence parallel to one side. the fencing around the outside of the enclosure must be barbed at the top to keep local thugs from stealing your farm animals. therefore the outside fencing costs $4 per linear foot while the inside fencing only costs $2 per linear foot. if the total area of the enclosure must be 500ft^2 find the dimensions that will minimize the total cost of fencing.

Explanation / Answer

Let the width of the field be y and let the length be 2x

Area = 2xy = 500, so xy = 250 and y = 250/x

Cost C = 4(4x + 2y) + 2y = 16x + 10y = 16x + 10(250/x) = 16x + 2500/x

dC/dx = 0 means 16 - (2500/x^2) = 0

x^2 = 2500/16 = 156.25, so x = 12.5

2x = 25 and y = 250/12.5 = 20

The dimensions of the field are: Length = 25 ft and Width = 20 ft.

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