a farmer wishes to fence off a rectangular field with 1,000 feet of fencing.if the long side of the field is along a stream, express the area of the field as a function of its width
width --- w
length --- l
l + 2w = 1000
l = 1000 - 2w
area = w l
= w(1000 - 2w)
To express the area of the field as a function of its width, let's assume the length of the field is L and the width is W.
We know that the perimeter of the rectangular field is given by the formula:
Perimeter = 2L + 2W
In this case, the perimeter is given as 1,000 feet. So we have:
2L + 2W = 1000
Simplifying the above equation, we get:
L + W = 500
Now, since the long side of the field is along a stream, the fence will run along the length twice and the width once. So we can express the perimeter as:
Perimeter = 2L + W
Since the total fencing available is 1,000 feet, we can write:
2L + W = 1000
Solving the equation above for L, we get:
L = (1000 - W) / 2
Now we can calculate the area of the rectangular field:
Area = Length × Width
Substituting the value of length (L) from the equation above into the area formula, we have:
Area = [(1000 - W) / 2] × W
So, the area of the field is expressed as a function of its width (W) as:
Area(W) = (1000 - W)W / 2
This equation represents the area of the rectangular field in terms of its width.
To express the area of the field as a function of its width, let's first label the dimensions of the rectangular field.
Let's say the width of the field is "w" feet. Since the long side of the field is along a stream, it will have only three sides to fence: the two ends and the width. Hence, the length of the field will be "1000 - 2w" feet.
Now, to calculate the area of the field, we multiply its length by its width. Thus, the area (A) can be expressed as:
A = length × width
A = (1000 - 2w) × w
A = 1000w - 2w^2
Therefore, the area of the rectangular field can be expressed as a function of its width (w) as A = 1000w - 2w^2.