A gardener is putting a wire fence along the edge of his garden to keep animals from eating his plants.
-If he has 20 meters of fence, what is the largest rectangular area he can enclose?
-What quadratic function represents the area of the garden?
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Considering all rectangles with the same perimeter, the square encloses the greatest area.
Proof: Consider a square of dimensions x by x, the area of which is x^2. Adjusting the dimensions by adding a to one side and subtracting a from the other side results in an area of (x + a)(x - a) = x^2 - a^2. Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.
Given 20 meters of fence, the garden is 5 x 5.
*--Considering all rectangles with the same area, the square results in the smallest perimeter for a given area.
*--Given a fixed length of fence, the circle encloses the maximum area.
To find the largest rectangular area that the gardener can enclose with a 20-meter fence, we need to determine the dimensions of the rectangle.
Let's assume the length of the rectangle is L meters and the width is W meters.
Since the gardener has a total of 20 meters of fence available, the perimeter of the rectangle can be expressed as:
2L + 2W = 20
Simplifying this equation, we get:
L + W = 10
To find the largest area, we need to maximize the product of L and W. As the sum of L and W is fixed at 10, we can use the quadratic function to represent the area of the rectangle.
The area, A, is given by the product of the length and width:
A = L × W
To eliminate one variable, we can solve the equation L + W = 10 for L, which gives:
L = 10 - W
Substituting this value of L into the area equation, we get:
A = (10 - W) × W
This quadratic function represents the area of the garden.
Alternatively, if you prefer to work with the equation in standard quadratic form (ax² + bx + c), we can expand the earlier equation:
A = 10W - W²
So, the quadratic function representing the area of the garden is A = 10W - W².
To find the largest rectangular area that the gardener can enclose using the wire fence, we need to consider the shape with the maximum area for a given amount of fence.
We know that a rectangle has four sides, with opposite sides being equal in length. Let's assume the length of the rectangle is L and the width is W. Since we have 20 meters of fence, the perimeter of the rectangle will be:
Perimeter = 2L + 2W = 20
We can rearrange this equation to solve for W:
W = (20 - 2L)/2
W = 10 - L
The area of a rectangle is given by the formula: Area = Length × Width. Substituting the equation for W into the area formula, we get:
Area = L × (10 - L)
To find the largest possible area, we need to express the area in terms of a quadratic function. Expanding the equation, we have:
Area = 10L - L^2
This quadratic function represents the area of the garden in terms of its length.
To find the maximum area, we can analyze the graph of the quadratic function. The maximum area corresponds to the vertex of the parabola. In this case, the parabola opens downward since the coefficient of the quadratic term (-1) is negative. This means that the vertex represents the maximum point.
To find the x-coordinate of the vertex, we can use the formula: x = -b/2a, where a and b are the coefficients of the quadratic function. In our case, a = -1 and b = 10.
x = -10/(2(-1))
x = -10/-2
x = 5
Therefore, the length L should be 5 meters to achieve the largest rectangular area. Substituting this value back into the equation for W, we can find the width:
W = 10 - L = 10 - 5 = 5 meters
Thus, the largest rectangular area the gardener can enclose with 20 meters of fence is a square with sides measuring 5 meters each. The area of this garden would be 5 meters × 5 meters = 25 square meters.
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Question 812207: A gardener is putting a wire fence along the edge of his garden to keep animals from eating his plants. If he has 20 meters of fence what is the largest rectangular area he can enclose?
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A gardener is putting a wire fence along the edge of his garden to keep animals from eating his plants. If he has 20 meters of fence what is the largest rectangular area he can enclose?
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let x=length