A home owner is planning the design of his rectangular vegetable garden as shown in the diagram. He plans to put wire fencing around each patch of vegetables to separate them and keep out the local deer. He has purchased 126 m of fencing. What dimensions should he use to maximize the area of the garden?
Let's denote the length of the rectangle as L and the width as W.
To maximize the area, we need to find the dimensions that will give us the largest possible area.
The perimeter of the rectangle is given by the equation:
2L + 2W = 126 m
We can rearrange this equation to solve for L:
2L = 126 m - 2W
L = 63 m - W
The area of the rectangle is given by the equation:
A = L * W
Substituting the expression for L from the earlier equation, we have:
A = (63 m - W) * W
Expanding the equation, we get:
A = 63W - W^2
We can now find the value of W that will maximize the area by finding the vertex of this quadratic equation. The x-coordinate of the vertex can be found using the formula:
x = -b / 2a
In this case, a = -1 and b = 63, so the x-coordinate of the vertex is:
W = -63 / (2 * -1) = 31.5
Since the width cannot be negative, we can discard negative values. Therefore, the width of the rectangle should be 31.5 m.
Substituting this value back into the equation for L, we have:
L = 63 m - 31.5 m = 31.5 m
So the dimensions that the homeowner should use to maximize the area of the garden are 31.5 m by 31.5 m.