A landscape architect plans to enclose a 2,500 square feet rectangular region in a botanical garden. She will use shrubs costing 18 dollar per foot along three sides and fencing costing 32 dollars per foot along the fourth side. Find the dimensions of the botanical garden that will minimize the total cost. Follow the steps:

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.

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