Find the dimensions of the garden that minimize the cost

Question

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $30/ft and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 22 square feet, find the dimensions of the garden that minimize the cost.

Length of side with bricks x=

Length of adjacent side y=

Answer 1

Let x = length of brick wall

 

      y = length of a side of fence that is perpendicular to the wall

 

Then, total length of fence = x + 2y

 

Since xy = 22, y = 22/x.  So, length of fence = x + 44/x

 

C(x) = total cost

 

        = 30x + 10(x + 44/x)

 

        = 40x + 440/x, x > 0

 

C'(x) = 40 - 440/x² = (40x²-440)/x²

 

C'(x) = 0 when 40x² = 440

 

                           x² = 11   So, x = √11

 

When 0<x<√11, C'(x) < 0, so C(x) is decreasing

When x > √11, C'(x) > 0, so C(x) is increasing

 

Therefore, the cost is minimized when x = √11 ft

                                                          y = 22/√11 = 2√11 ft

                                                                             

 

 

Answer 2

Hi Sydney,

 

the area is 22 square feet --> XY = 22 --> Y = 22/X

 

The brick side is $30/foot and the other three sides are $10/foot--> Cost = 30X + (3)(10)Y

 

Substitute--> Cost = 30X + (30)(22)/X

 

C = 30X + 660/X

 

C' = 30 + (X - 660)/X^2

 

Set this equal to zero to find maximum/minimum--> 0 = 30 + (X - 660)/X^2

 

-30 = (X - 660)/X^2

 

cross multiply--> -30X^2 = X - 660

 

-30X^2 - X + 660 = 0

 

Multiply by -1--> 30X^2 + X - 660 = 0

 

quadratic equation where a = 30   b = 1   c = -660

 

X = (-1 +/- sq root (1 + 79200))/60

 

X = (-1 +/- 281.4)/60

 

X = 4.67   or X = a negative number which doesn't make sense.

 

Then Y = 4.71

 

Then the total cost  is $281.40.  Try plugging in X values a little more and little less than 4.67 to confirm that this is the minimum cost.

 

Have fun.

 

Dan