A fence must be built to enclose a rectangular area of ft . Fencing material costs $ #!ß !!! $ # per foot for the two sides facing north and south and $ per foot for the other two sides. Find the ' cost of the least expensive fence

the numbers are missing, so I looked for a similar question:

A fence must be built to enclose a rectangular area of 20,000 ft². Fencing material costs $2.50 per foot for the two sides facing north and south and $3.20 per foot for the other two sides. Find the cost of the least expensive fence.

we must first write down the area and perimeter equations, where X = sides facing north and south, and Y = sides facing east and west.

area ⇒ XY = 20000

perimeter ⇒ P = 2X + 2Y

X = 20,000 / Y

now we replace X in the second equation:

P = 2 · 20000/Y + 2Y

P = 40000/Y + 2Y

now we write down our cost equation:

cost = C(y) = (2.50 · 40000/Y) + (3.20 · 2Y)

cost = C(y) = 6.4Y + 100000/Y

to determine the minimum cost we must find the derivative of the cost equation:

C'(y) = 6.4 - 100000/Y²

now, C'(y) = 0

0 = 6.4 - 100000/Y²

100000/Y² = 6.4

Y² = 100000/6.4 = 15,625

Y = √15,625 = 125

this means that Y = 125 feet

X = 20000/125 = 160 feet

perimeter = (2 · 160) + (2 · 125) = 320 + 250 = 570 feet

cost of fence = (320 · $2.50) + (250 · $3.20) = $1,600

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