Bus. Calc dimensions

Question

For reasons too complicated to explain, I need to create a rectangular orchid garden with an area of exactly 169 sq. ft. abutting my house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $4 per foot, and the fencing for the east and west sides costs $2 per foot. What are the dimensions of the orchid garden with the least expensive fence?

Answer 1

169=xy

 

2(4x)+2(4y)=C(x)

 

169/x=y

 

8x + 8(169/x)=C(x)

 

Take 1st derivative. 

 

8 - 1352/x²=C'(x)

 

Set C'(x)=0

 

and you get 

 

-1352=-8x²

 

Solve by dividng by -8 to get x²=169.  Take square root and you will get x = 13. Plug 13 into 169=13y.  You get y = 13. 

 

Your dimensions should be 13 x 13.