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? p=7x+9y and A= 2x+4y
To maximize the area of the garden, we need to find the dimensions of the garden that will yield the maximum value for A.
Given that the perimeter of the garden is 126 m, we can write an equation for the perimeter based on the given dimensions of the garden: 2x + 2y = 126.
Simplifying this equation, we get: x + y = 63.
Now let's solve this equation for either x or y in terms of the other variable. For convenience, we'll solve for y: y = 63 - x.
Substitute this expression for y in the formula for A: A = 2x + 4(63 - x).
Simplifying further, we get: A = 2x + 252 - 4x.
Combining like terms, we get: A = -2x + 252.
To maximize the area, we need to find the maximum value of A. Thus, we differentiate A with respect to x and set it equal to 0:
dA/dx = -2 + 0 = -2.
Setting -2x + 252 = 0 and solving for x, we get: x = 126.
Substituting this value of x back into the equation y = 63 - x, we find y = 63 - 126 = -63. However, since negative dimensions are not realistic for a garden, we discard this solution.
Therefore, there is no valid solution that maximizes the area of the garden.