Afarmer has 1600 yards of fencing to enclose a rectangular garden. express the area a of the rectangle as a function of the width x of the rectangle. what is the domain of a?

Let the length of the rectangular garden = l yards

And the width of the garden = w yards

Farmer has the fence in length = 1600 yards to cover all the sides of the garden

Therefore, Perimeter of the rectangular garden = 1600 yards

Since, perimeter of the rectangular garden is given by the expression,

Perimeter = 2(l + w)

By substituting the values in the expression,

1600 = 2(l + x)

l + x = 800

l = 800 - x (1)

Since, area of the rectangle = Length × Width

                                                = l × w

                                                = lx

Therefore, by substituting the value of 'l' in terms of 'x' in from equation (1),

Area of the rectangular garden = (800 - x)x

If the area of the garden is represented by by the function A(x),

A(x) = (800 - x)x

A(x) = 800x - x²

Since, Area can not be negative or zero,

Domain of the function will be 0 < x < 800

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A = (800 yd - x)x, [0, 800]

Step-by-step explanation:

The farmer fences in this rectangular garden using 1600 yds of fencing.  Using the formula for the perimeter of a rectangle,

P = 2x + 2L, where x is the width and L is the length.  Here P = 1600 yd.

Solving for L:  1600 yd = 2x + 2(L) →  800 yd = x + L  →  L = 800 yd - x.

The area of the rectangle is A = L·x.  Subbing (800 yd - x) for L, we get:

A = (800 yd - x)x.  This is the desired formula for the area of the rectangle as a function of x alone.  Neither length nor width can be negative, so the domain of this function A is x ≤ 800 yd.

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