Rectangular plot of land, $5 foot, sides being $2 foot, cost $13,000

Question

A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for $5 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $13,000?

Answer 1

Area = xy

2x(5) + 2y(2) = 13,000

5x + 2y = 6500

2y = 6500 -5x

y = 3250 -5x/2

Area = A = xy = x(3250-5x/2) = 3250x -(5/2)x^2

take the derivative and set equal to zero

A' = 3250 -5x = 0

5x =3250

x = 650 feet of $5 heavy duty fencing

y =3250-(5/2)(650) = 3250 - 5(325) = 5(325) = 1625 feet of $2 standard fencing

max Area with $13,000 is 650(1625) = 105,625 square feet

standard fence is 2 1/2 times cheaper, so you use 2 1/2 times as much standard as heavy duty

1625 = 2.5(650)

10x + 4y = 13000

5x + 2y = 6500

5x + 2(5x/2) = 6500

10x = 6500

x = 6500/10 = 650

y = 2.5(650) = 1625