A rancher has 220 feet of fencing to enclose a rectangular corral. Find the dimensions of the rectangle that maximize the enclosed area, using all 220 feet of fencing. Then find the maximum area.
A square will provide the largest area.
220/4 = 55 feet on each side
A = 55 * 55
A = ?
but its a rectangle
A square is a rectangle.
buuutttttt the sides will not be all the same length
To find the dimensions of the rectangle that maximize the enclosed area, we can use the quadratic formula. Let's assume the length of the rectangle is x and the width is y.
First, let's find an equation relating the perimeter of the rectangle to the given fencing. The perimeter is equal to the sum of all sides of the rectangle, which gives us:
2x + 2y = 220
We can simplify the equation to:
x + y = 110 ... (1)
Next, let's find an equation relating the area of the rectangle to the dimensions. The area is the product of the length and width, which gives us:
A = xy
Now, we need to express one variable in terms of the other. From equation (1), we can solve for y:
y = 110 - x
Substituting this into the area equation, we get:
A = x(110 - x)
Now, we have a quadratic equation for the area in terms of one variable (x). To find the maximum area, we need to determine the x-value at the vertex of the parabolic graph. The vertex of a quadratic equation in the form ax^2 + bx + c is given by:
x = -b / (2a)
In our case, a = -1, b = 110, and c = 0. Substituting these values into the formula, we have:
x = -110 / (2(-1))
x = 55
So, the length of the rectangle is x = 55 feet. Substituting this back into equation (1), we can find the width:
55 + y = 110
y = 110 - 55
y = 55
Therefore, the width of the rectangle is y = 55 feet as well.
Finally, let's find the maximum area by substituting the values of x and y into the area equation:
A = x * y
A = 55 * 55
A = 3025 square feet
So, the dimensions of the rectangle that maximize the enclosed area using all 220 feet of fencing are 55 feet by 55 feet, and the maximum area is 3025 square feet.