Question

A gardener with 1000 m1000 m of available fencing wishes to enclose a rectangular field and then divide it into two plots with a fence parallel to one of the sides as shown in the figure.

What is the largest area that can be enclosed?

Answer 1

To find the largest enclosed area using 1000 meters of fencing, the dimensions of the rectangular field need to be maximized. By expressing the area as a quadratic function and finding its vertex, we can determine the dimensions that will result in the largest area. The maximum area that can be enclosed is 125,000 square meters.

Step-by-step explanation:

To find the largest area that can be enclosed, we need to maximize the area of the rectangular field. Let's assume the length of the field is x meters and the width is y meters. Since the field is divided into two plots with a fence parallel to one of the sides, the total length of fencing used will be 2x + y. And we know that the total length of fencing available is 1000 meters.

So we have the equation:

2x + y = 1000

Now, to find the largest area, we need to express the area in terms of either x or y and then maximize it. Let's express the area in terms of x:

Area = x*y

Solving the first equation for y, we get:

y = 1000 - 2x

Substituting this value of y in the area equation:

Area = x*(1000 - 2x)

This equation represents a quadratic function in standard form, where the coefficient of x^2 is -2. Since the coefficient of x^2 is negative, the graph of the function will be an upside-down parabola. The maximum area will occur at the vertex of the parabola.

The x-coordinate of the vertex can be found using the formula: x = -b/(2a), where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = -2 and b = 1000.

Substituting these values, we get:

x = -1000/(2*(-2)) = 250

Now, substituting this value of x in the equation for y, we get:

y = 1000 - 2*250 = 500

So the dimensions of the rectangular field that maximize the area are 250 meters by 500 meters. And the largest area that can be enclosed is:

Area = 250*500 = 125,000 square meters

Answer 2

The largest area that the gardener can enclosed with 1,000 m of fencing is about 41,666 square meters

The steps used to obtain the largest area that can be enclosed with the fencing available are as follows;

Let x represent the length of the fencing parallel to one of the sides, we get;

Length of the fencing around the perimeter of the garding = 2 × Width of the garden + 2 × Length of the garden

Width = Length of the parallel fence, x

Length = (1000 - Length of parallel fence - 2 × Width)/2

Length = (1000 - x - 2·x)/2

(1000 - x - 2·x)/2 = 500 - 3·x/2

Area of the garden = Length × Width

Area, A(x) = x × (500 - 3·x/2)

x × (500 - 3·x/2) = 500·x - 3·x²/2

A(x) = 500·x - 3·x²/2

The function is a quadratic function and the largest area can be obtained when the x-value is; -500/(2 × -3/2) = 500/3 meters

500/3 = 166 2/3

Therefore, the maximum area, A(x)
`_(max)` = 500 m × (500/3) m - 3×(500/3 m)²/2

A(x)
`_(max)` ≈ 41,666.67 m²