Find the length and width (with W<=L ) of the enclosure that is most economical to construct.

Question

A fence is to be built to enclose a rectangular area of 240 square feet.The fence along three sides is to be made of material that costs 5 dollars per foot,and the material for the fourth side costs 12 dollars per foot.

Answer 1

4 sides, 3 cost $5 per foot5

4th costs $12 per foot

LW = 240

L=240/W

Cost = C = 5(2W) + 5L +12L

= 10W + 17(240/W)

=10W + 17(240)/W

take the derivative and set = 0

C' = 10 -17(240)/W^2 = 0

multiply by W^2

10W^2 -17(240) = 0

W^2 = 17(24)

W = sqr(17(24)) = about 20.20 ft

L = 240/sqr(17(24)) = about 11.88 ft

20.20 by 11.88 feet minimizes Cost

Answer 2

W ≤ L

WL = 240

c = 5, c' = 12

C( W, L ) = Some function, f, of W and L [ f( W, L ) ]

'Most economical to construct' seems to communicate the lowest cost possible to achieve the fence project.

Strangely, it is when the 2 L walls are at a cost of $5/ft and 1 W wall is at $5/ft and the other is $12/ft.

Because of the postulation of W ≤ L, the sum of the two W walls should be less than the sum of the two L walls.

Thus,

C = 5 ( 2L + W ) + 12 W

C = 10 L + 17 W

W = 240 / L

C = 10 L + 17 ( 240 / L )

C = 10 L + 4080 / L

C' = 0 = 10 - 4080 / L²

After some algebra and equation manipulation,

10 L² = 4080

∴ L ≈ 20.20 ft

W = 240 / L

∴ W ≈ 11.88 ft

Note: If you attempted the cost the other way, you'd get the same numbers, just on opposite variables and conflict with the given W ≤ L prompt postulation.

I hope this helps! Please message me in the comments if you have any questions, comments, or concerns!