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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 2100 m of wire at your​ disposal, what is the largest area you can​ enclose, and what are its​ dimensions?

Respuesta :

pedrocarratore

Answer:

525 x 1,050

A = 551,250 m²

Step-by-step explanation:

Let 'L' be the length parallel to the river and 'S' be the length of each of the other two sides.

The length of  the three sides is given by:

[tex]2S+L=2,100\\ L=2100-2S[/tex]

The area of the rectangular plot is given by:

[tex]A=S*L\\A=S(2100-2S)\\A=2100 S -2S^2[/tex]

The value of 'S' for which the area's derivate is zero, yields the maximum total area:

[tex]\frac{dA(S)}{dS}=\frac{d(2100 S -2S^2)}{dS}\\0= 2100 - 4S\\S=525[/tex]

Solving for 'L':

[tex]L=2100-(2*525)\\L=1,050[/tex]

The largest area enclosed is given by dimension of 525 m x 1,050 and is:

[tex]A = 525*1,025\\A=551,250\ m^2[/tex]

raphealnwobi

The maximum area is 275625 m² with a length of 525 m and width of 525 m.

A rectangle is a quadrilateral in which opposite sides are parallel and equal to each other.

The perimeter of a rectangle is:

Perimeter = 2(length + breadth)

Let x represent the length and y the breadth, hence:

Perimeter = 2(length + breadth)

2100 = 2(x + y)

x + y = 1050

y = 1050 - x

Area (A) = length * breadth = xy

A = x(1050 - x)

A = 1050x - x²

Maximum area is at dA/dx = 0, hence:

1050 - 2x = 0

2x = 1050

x = 525 m

y = 1050 - x = 1050 - 525 = 525

Area = 525 * 525 = 275625 m²

The maximum area is 275625 m² with a length of 525 m and width of 525 m.

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