The Maximum Garden Problem. A farmer has 230 ft
of fence to enclose a rectangular garden. What is the
largest garden area that can be enclosed with the 230 ft
of fence?
The quick answer is that for a given perimeter, the largest area possible is a square. Thus each side equals 1/4 of the perimeter = 230/4 = 57.5'
Area = 57.5² sq.ft.
= 3306.25 sq.ft.
The mathematical method is to calculate the area in terms of the permimeter and one of the sides (x). Differentiate with respect to x and equate the derivative to zero to find the maximum area.
Length of one side = x
Length of the other side (of rectangle)
= 230/2-x
Area,
A(x)
= x(115-x)
= 115x -x²
A'(x) = 115-2x
x=57.5
A"(x) = -2 <0 so x=57.5 is a maximum.
Proceed to calculate the area.
To find the largest garden area that can be enclosed with 230 ft of fence, we need to determine the dimensions of the rectangular garden that will maximize the area.
Let's assume the length of the garden is 'l' and the width is 'w'.
To enclose the garden with the given fence, we use the formula for the perimeter of a rectangle: P = 2(l + w).
Given the perimeter as 230 ft, we can write the equation as follows:
230 = 2(l + w)
Now, let's solve this equation for one of the variables (l or w) in terms of the other. For example, we can solve for 'l' in terms of 'w' by subtracting 'w' from both sides:
230 - 2w = 2l
115 - w = l
Now, we can express the area of the garden (A) in terms of 'w':
A = lw
A = w(115 - w)
To find the largest garden area, we need to find the value of 'w' that maximizes this equation. This can be done by finding the maximum point of the quadratic equation.
To find the maximum point of a quadratic equation, we need to determine the vertex. The vertex form equation is given as:
A = a(x-h)² + k, where (h, k) is the vertex.
In our case, the equation is A = w(115 - w). To put it in vertex form, we can rewrite it as:
A = -w² + 115w
Comparing this with the vertex form equation, we can identify 'a' as -1, 'h' as 57.5 (half of 115), and 'k' as 3322.5 (calculated by substituting h and k into the equation: A = -(-57.5)² + 115(-57.5) = 3322.5).
The vertex form equation becomes:
A = -(w - 57.5)² + 3322.5
Now, we know that the value of 'w' that maximizes the equation is the x-coordinate of the vertex. In this case, w = 57.5.
Substituting this back into our equation for 'l':
l = 115 - w
l = 115 - 57.5
l = 57.5
So, the dimensions of the rectangular garden that would enclose the largest area with 230 ft of fence are 57.5 ft by 57.5 ft.
To find the largest garden area, substitute the values of 'l' and 'w' into the area equation:
A = l * w
A = 57.5 * 57.5
A = 3306.25 ft²
Therefore, the largest garden area that can be enclosed with 230 ft of fence is 3306.25 square feet.