a Farmer want to put up a fence with a total perimeter of 250 ft. It is the shape of a rectangle but there is a river on one side. What is the maximum area of feet that the fence can be?
What is the max. Area of the three sides
So if the perimeter is 250 feet, draw a diagram. of three sides of a rectangle with a little river. Then label the two corresponding sides 'x' and the other side without a partner 250-x because the total P is 250 and you already used 'x' twice on the other sides of the fence. Then, because area is A=LW, substitute in the sides: A=(250-2x)x
To find the maximum area that the fence can enclose, we need to understand the constraints given in the problem. We know that the shape of the fence is a rectangle, but one side of the rectangle is adjacent to a river. This means that the river forms one side of the rectangle and does not require fencing.
Let's denote the dimensions of the rectangle as follows:
Length (adjacent to the river) = L
Width (opposite side of the river) = W
From the given information, we know that the total perimeter of the fence is 250 ft.
Perimeter of a rectangle = 2 * (Length + Width)
Using this formula, we can write the equation:
250 = 2 * (L + W)
Now, we want to determine the maximum area that the fence can enclose, which is the product of its length and width:
Area of a rectangle = Length * Width
We need to express the area in terms of only one variable. For this, we can rearrange the perimeter equation to solve for L:
250 = 2 * (L + W)
125 = L + W
L = 125 - W
Substituting this value of L in the area equation:
Area = (125 - W) * W = 125W - W^2
To find the maximum area, we need to find the maximum value of this quadratic function.
To find the maximum point of a quadratic equation, we can use the vertex formula:
x = -b / (2a)
In our equation, a = -1 and b = 125. Substituting these values:
W = -125 / (2 * -1)
W = -125 / (-2)
W = 62.5
Since width cannot be negative, the width is 62.5ft. Similarly, the length can be found using the perimeter equation:
125 = L + 62.5
L = 62.5
Therefore, the maximum area that the fence can enclose is:
Area = Length * Width
Area = 62.5ft * 62.5ft
Area = 3906.25 square feet
So, the maximum area that the fence can enclose is 3906.25 square feet.