Question

a rectangular area of 24000 square feet is to be fenced on all four sides. fencing east and west sides cost $10 per foot fencing for the other two sides cost $20 per foot. what s the cost of least expensive

Answer 1

C(min) = 8763.4 $

Step-by-step explanation:

Area = 24000 ft²

In a rectangle, A = x * y x is the base side and y the height side

Cost of low and upper side (x) is 20 $ per foot

Cost of east and west sides ( y) is 10 $per ft

total cost is:

C(r) = 2 * 20* x + 2* 10* y

from A = x * y y = A / x y = 24000 / x

And by substitution in C(r) we get:

C(x) = 2 * 20* x + 2* 10* 24000 / x

C(x) = 40 * x + 480000 / x

Tacking derivatives on both sides of the equation:

C´(x) = 40 - 480000 / x²

C´(x) = 0 40 - 480000/x² = 0

40* x² - 480000 = 0

x² = 480000 / 40

x² = 12000

x = √ 12000 = 109,54 ft

and y = 24000 / 109,54

y = 219,09 ft

Chequing for second derivative

C´´(x) = 480000 / x⁴ is always positive so we have a minimum of C at the value x = 109,54

Minimum cost C (min) = 40* 109,54 + 20 * 219,09

C(min) = 4381.6 + 4381.8

C(min) = 8763.4 $