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 1500 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions

Step-by-step explanation:

Let x be the length and y be the width of the rectangular plot.

The plot is bounded on one side by a river and on the other three sides by a single-strand electric fence. It means,

x+2y = 1500

x = 1500 - 2y ....(1)

We know that the area of a rectangular plot is given by :

A = xy ....(2)

Put the value of x from equation (1) in (2)

[tex]A=(1500-2y)y\\\\A=1500y-2y^2[/tex] .....(3)

For largest area, differentiate above area equation wrt y.

[tex]\dfrac{dA}{dy}=\dfrac{d}{dy}(1500y-2y^2)\\\\=1500-4y\\\\\text{Put}\ \dfrac{dA}{dy}=0\\\\1500-4y=0\\\\y=\dfrac{1500}{4}\\\\=375[/tex]

Put the value of y in equation (1).

x = 1500-2(375)

= 750 m

Put the value of y in equation (3).

[tex]A =1500(375)-2(375)^2\\A=281250\ m^2[/tex]

Hence, the largest area is 281250 m² and its dimensions are 750 m and 375 m.

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