Jimmy is wanting to fence a rectangular area of his land along Elm Creek. He will only need to fence 3 sides. If Jimmy has 2000 ft of fencing and wants to enclose the largest possible area, determine the dimensions of the rectangular plot. Hint: Write an Area function in terms of x, the width of the lot. Draw a diagram and label. Also, draw a rough sketch of the Area function.
Closest to a square is largest area.
2000/3 = ?
the largest area is when the fence is divided equally among lengths and widths.
so, 1000 for the length, 1000 for the two widths.
The largest area is 1000x500
using the hint,
A = w(2000-2w)
To determine the dimensions of the rectangular plot, we need to maximize the enclosed area given the constraint of having only 2000 ft of fencing.
Let's start by assuming the width of the lot is "x" feet. In a rectangular plot, the width will be one of the sides that needs to be fenced, and the other two sides will be the length of the plot.
Since we have 2000 ft of fencing and only need to fence three sides, the equation for the perimeter of the plot can be written as:
Perimeter = 2(width) + length + length
2000 = 2x + 2(length)
Simplifying the equation further:
2000 = 2x + 2length
1000 = x + length
Now, we can write the equation for the area of the plot, which is length multiplied by width:
Area = length * width
To express the area in terms of x, we need to substitute length in the area equation with (1000 - x), based on the previous equation.
Now the area equation becomes:
Area = x * (1000 - x)
To maximize the area, we can take the derivative of the area equation with respect to x, set it to zero, and solve for x. This will give us the width of the lot that maximizes the area.
The area function is graphed as a quadratic function with a maximum point, giving us a parabolic curve. The curve opens downwards since the coefficient of the x^2 term is negative (-1).
To find the maximum point, we need the x-coordinate where the slope of the tangent line is zero.