This is section 3.8 problem 16: 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 cost function is given by:

Total Cost = 32w + 18 (w + 2l) = 50w + 36l

Where,

w: width

l: long

The area is given by:

Area = w * l = 2500

We cleared l:

l = 2500 / w

We leave the cost function in terms of a variable:

Cost = 50w + 36 (2500 / w) = 50w + 90000 / w

We now look for the derivative of the function:

C ' (w) = 50 - 90000 / w² = 0

50 = 90000 / w²

50w² = 90000

w² = 90000/50

w² = root (90000/50)

w = 42.43 feet

We look for the other dimension:

l = 2500 / (42.43)

l = 58.92 feet

Minimum cost:

C (42.43) = 50 * (42.43) + 36 * (2500 / 42.43) ≈ $ 4242.64

Answer:

The dimensions of the botanical garden that will minimize the total cost are:

w = 42.43 feet

l = 58.92 feet

The cost is:

$ 4242.64