Question

a rectangular field is to be enclosed on four sides with a fence. fencing costs 5$ per foot two opposite sides, and 6$ per foot for the other two sides. find the dimensions of the feild of area 710ft

Answer 1

Length of cheaper side = x = 29.21 ft

Length of expensive side = y = 24.31 ft

Explanation:

Let

Length of cheaper side = x

Length of expensive side = y

Total Cost = 5(x + x) + 6(y+y) =

Total Cost = 5(2x) + 6(2y)

Total Cost = 10x + 12y

we also know that

xy = 710ft square

so

y = 710 ft sq. / x

Putting this in total cost equation

Total Cost = 10x + 12( 710 ft sq. / x )

f(x) = 10x + ( 8,520 ft sq. / x ) ( cost = f(x)

f'(x) = 10 - ( 8,520 ft / x^2 ) (x>0)

0 + ( 8,520 ft / x^2 ) = 10

8,520 ft / x^2 = 10

x^2 = 8,520 ft / 10

(x^2)^1/2 = (8,520 ft / 10)^1/2

x = 92.3 / 3.16

x = 29.21 ft

As given

xy = 710ft square

Putting value of x in above equation

29.21 y = 710

y = 710/29.21

y = 24.31 ft