Answer:
The dimensions of the botanical garden that minimizes the cost of the rectangular botanical garden is 42.43 ft. × 58.93 ft.
Explanation:
Let the dimensions of the botanical garden that minimizes cost be x and y respectively.
xy = 2500
The cost function is then
C(x, y) = 32x + 18x + 18y + 18y = 50x + 36y
(Since, one side is fenced using $32 per foot material while the other 3 remaining sides are fenced using $18 per foot shrub)
So, the function to.be minimized is the cost function
C(x,y) = 50x + 36y
subject to the constraint
xy = 2500
Using the substitution method, the constraint equation becomes
xy = 2500
y = (2500/x)
Substituting this into the cost function
C(x,y) = 50x + 36y
C(x) = 50x + 36(2500/x)
C(x) = 50x + (90,000/x)
So, to minimize the function now,
At minimum cost, (dC/dx) = 0 and (d²C/dx²) > 0.
C(x) = 50x + (90,000/x)
(dC/dx) = 50 - (90,000/x²)
At minimum Cost
50 - (90,000/x²) = 0
50 = (90,000/x²)
50x² = 90,000
x² = 1800
x = 42.43 ft.
To check if it indeed corresponds to minimum point,
(d²C/dx²) = (180,000/x³)
Since x cannot be negative, the second derivative, (d²C/dx²) = +ve > 0, hence, this point really corresponds to a minimum point for the cost function.
At minimum cost, x = 42.43 ft
xy = 2500
y = (2500/x) = (2500/42.43) = 58.93 ft.
Hence, the dimensions of the botanical garden that minimizes the cost of the rectangular botanical garden is 42.43 ft. × 58.93 ft.
Hope this Helps!!!
