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Question

This is an optimization problem

 

doug is planning a rectangular garden with roses along the front and a hedge along the other three sides. Along the front he wants an opening which is 5 ft long. The garden is to be 4500ft^2. if the hedge cost $10 per foot and the roses cost $15 per foot what dimensions shoudl Doug use to minimize the total cost for roses and hedge?

Answer 1

Let X = length of front and back of rectangle

Let Y = length of sides of rectangle 

 

Total cost = Side  + Side  +  Back + Front

              = 10*Y + 10*Y + 10*X + 15*(X-5)

              = 20*Y + 25*X - 75 

 

Area = 4500  ==>  XY = 4500  ==>  Y = 4500/X

 

C(X) = 20*(4500/X) + 25*X - 75 = 25*X -75 + 90000*X^-1

 

C'(X) = 25 - 90000*X^-2 = 25 - 90000/X²

 

C'(X) = 0 when 90000/X² = 25

X² = 90000/25 = 3600

X = +√3600 = 60 

 

C''(X) = 180000/X³ 

C''(60) > 0 so X = 60 is a minimum 

 

Y = 4500/X = 4500/60 = 75 

 

The garden should be 60 ft in the front and back and 75 ft on the sides