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 say the length of the rectangular vegetable garden is L and the width is W.
To maximize the area of the garden, we need to find the optimal dimensions where L * W is the largest.
The perimeter of the garden is given by 2L + 2W, and since each patch of vegetables will have wire fencing around it, we can equate the perimeter to the total length of wire fencing purchased:
2L + 2W = 126
Simplifying the equation, we get:
L + W = 63
Solving for L in terms of W, we get:
L = 63 - W
To maximize the area, we need to differentiate it with respect to one variable and set the derivative equal to zero:
Area = L * W
d(Area)/dW = (63 - W) * dW/dW + W * d(63 - W)/dW = 0
63 - 2W = 0
2W = 63
W = 31.5
Substituting this value of W back into the equation L + W = 63, we find:
L + 31.5 = 63
L = 63 - 31.5
L = 31.5
Therefore, the dimensions he should use to maximize the area of the garden are 31.5 m by 31.5 m.