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A farmer plans to fence a rectangular garden next to a river. There will be no fence along the river side. Draw a picture of this situation. The pasture will contain 405,000 square meters. Find the dimensions of the garden that will require the least amount of fencing.

Respuesta :

Answer:

x  =  450 m

y =  900 m

P (min)  = 1800 m

Step-by-step explanation:

Let´s call   "x"  and  "y" de sides of the rectangular garden, y is the side parallel to the river ( that is it will be fenced only one )

Then:

2*x + y = P        ( perimeter of the area)

And   x * y  = 405000 m²

y = 405000/ x

And P as a function of x is:

P(x) = 2*x +  405000/x

Tacking derivatives relative to x on both sides of the equation.

P´(x)  = 2  - 405000/x²

P´(x) =  0      2*x²  - 405000 = 0

x²  = 202500

x = 450 m

And y =  405000 / 450

y = 900 m

And

P(min) = 2*450 + 900

P(min) = 1800 m

We know P is minimm at x =450 snce the second derivative of P

P´´(x) = 405000*2*x / x⁴   is always positive

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