400 feet of fencing is to be used to enclose four adjacent pieces of land. What dimensions will produce the largest area?
The square encloses the largest area for a given amountof fence, but I wonder what about the shapes and sizes of the pieces of land.
To find the dimensions that will produce the largest area, we need to consider the properties of a rectangle. Let's assume that the length of each of the four adjacent pieces of land is x and the width is y.
Since there are four adjacent pieces of land, the total perimeter of the rectangle formed by them will be equal to the amount of fencing used, which is 400 feet. The perimeter of a rectangle is given by P = 2x + 2y.
Therefore, we can write the equation:
2x + 2y = 400
Now, let's solve this equation for one of the variables so that we can express the other one in terms of it. Let's solve for y:
2y = 400 - 2x
y = (400 - 2x)/2
y = 200 - x
Next, we want to find the dimensions that produce the largest area. The area of a rectangle is given by A = x * y. Substituting the expression for y from above, we have:
A = x * (200 - x)
A = 200x - x^2
To find the dimensions that produce the largest area, we need to maximize this function. We can do that by finding the vertex of the parabola represented by this equation. The x-coordinate of the vertex can be found using the formula: x = -b/2a, where a = -1 and b = 200 in our equation.
x = -200/(2*(-1))
x = 100
So, the width of the rectangle (y) is:
y = 200 - x
y = 200 - 100
y = 100
Therefore, if 400 feet of fencing is used to enclose four adjacent pieces of land, the dimensions that will produce the largest area is a rectangle with a length (x) of 100 feet and a width (y) of 100 feet.