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A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a​ single-strand electric fence. With 2400 m of wire at your​ disposal, what is the largest area you can​ enclose, and what are its​ dimensions?

Respuesta :

Answer:

x = 600 m

y = 1200 m

Amax = 720000  m²

Step-by-step explanation:

Let call x the smaller side of the rectangular plot and y the largest ( we assume we have one y side bounded by a river: Then

A(p)  Area of the plot x*y

A(p) = x*y

And perimeter of the plot ( to be fenced ) is:

P(p)  = 2*x + y = 2400     ⇒  y  = 2400 - 2*x

Area of rectangular plot as function of x:

A(x) = x * ( 2400 - 2x )

Taking derivatives on both sides of the equation

A´(x) = ( 2400 - 2x ) + (-2) *x    ⇒  A´(x) = ( 2400 - 2x ) - 2x

A´(x) = 0      ⇒   2400 - 4x = 0    ⇒  4x  = 2400  

x = 600 m

And y =  2400 - 2*x

y = 2400 - 1200

y = 1200 m

And the largest enclosed area is  Amax = 1200*600

Amax = 720000 m²

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