A farmer wants to build a rectangular pen which will be bounded on one side by a river and on the other three sides by a single-strand electric fence. If the farmer has 40 meters of wire to use, what is the largest area that the farmer can enclose?

The area of the pen is the product of its dimensions

The largest area the farmer can enclose is 200 square meters

The perimeter of the pen is given as:

Because one side of the fence is covered by a river, the perimeter would be:

Make y the subject

The area of the pen is:

Substitute

Expand

Differentiate

Set to 0

Rewrite as:

Divide through by 4

Substitute 10 for x in

Hence, the largest area the farmer can enclose is 200 square meters

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A) Dimensions;

Length = 20 m and width = 10 m

B) A_max = 200 m²

Step-by-step explanation:

Let x and y represent width and length respectively.

He has 40 metres to use and he wants to enclose 3 sides.

Thus;

2x + y = 40 - - - - (eq 1)

Area of a rectangle = length x width

Thus;

A = xy - - - (eq 2)

From equation 1;

Y = 40 - 2x

Plugging this for y in eq 2;

A = x(40 - 2x)

A = 40x - 2x²

The parabola opens downwards and so the x-value of the maximum point is;

x = -b/2a

Thus;

x = -40/2(-2)

x = 10 m

Put 10 for x in eq 1 to get;

2(10) + y = 40

20 + y = 40

y = 40 - 20

y = 20m

Thus, maximum area is;

A_max = 10 × 20

A_max = 200 m²

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