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I would like to create a rectangular orchid garden that abuts 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. If I have a budget of $120 for the project, what are the dimensions of the garden with the largest area I can enclose

Respuesta :

Answer:

x  =  15 ft

y  =  15 ft

A(max)  =  225 ft²

Step-by-step explanation:

Let call  "x "  and  " y "  sides of the rectangle x (side paralll to the northern  boundary, then:

A(r)  = x*y        and      4*x  +  2*2*y  =  120   or    4*x  + 4*y  = 120

4*x  +  4*y  = 120    ⇒  x  +  y  = 30    ⇒   y  =  30 - x

Area of the garden as a function of x is:

A(x)  =  x* ( 30 - x )    ⇒  A(x)  =  30*x  - x²

Taking derivatives on both sides of the equation

A´(x)  = 30  - 2*x

A´(x)  = 0   ⇒    30  - 2*x =  0

2*x  =  30

x = 30/2

x = 15 ft

And  y =  ( 30 - x )

y  =  15 ft

A(max)  =  15*15

A(max)  = 225 ft²

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