Question

A farmer wants to enclose 135000 square yards of land in a rectangular plot. The material used for the front side costs $2 per yard, the material for the other three sides costs $1 per yard. Determine the dimensions that will minimize cost.

Answer 1

let l = length and w = width

for area 135000 = lw or w = 135000/l (1)

perimeter P = 2l + 2w

substitute 1 from area formula

P = 2l + 135000/l

find dP/dl and thne let dP/dl = 0 and solve for l

this will given the minimum length and subsequently minimum width by sustituting into (1)

Answer 2

My calculation shows that the dimensions as follows will minimise cost:
Front and rear lengths both equal 300 yards.
Side lengths both equal 450 yards. The method follows.

Answer 3

Let the front and rear lengths equal a yards each.
Let the side lengths equal b yards each.
Then a * b = 135000
and b = 135000/a
Let cost = C
C = 2a + a + 2b = 3a + 2b
Substituting for b:
C = 3a + 2*(135000/a) = 3a + 270000/a
dC/dt = 3 - (270000/a^2)
Put the differential coefficient = 0 to solve for a minimum:
3 - (270000/a^2) = 0
From which a^2 = 270000/3 = 90000
a = 300 yards
by substitution in a * b = 135000 the value of b = 450 yards