Hi-
I have no idea how to start this....can anyone help me out?:
A gardener wants to use 130 feet of fencing to enclose a rectangular plot divided into two plots. What is the largest possible area for such a garden? Draw a picture and label the exact dimensions of the fencing used.
Considering all rectangles with a given perimeter, which one encloses the largest area?
The traditional calculus approach would be as follows.
Letting P equal the given perimeter and "x" the short side of the rectangle, we can write for the area A = x(P - 2x)/2 = Px/2 - x^2.
Taking the first derivitive and setting equal to zero, dA/dx = P/2 - 2x = 0, x becomes P/4.
With x = P/4, all four sides are equal making the rectangle a square.
.....The short side is P/4.
.....The long side is (P - 2(P/4))/2 = P/4.
Therefore, it can be unequivicably stated that of all possible rectangles with a given perimeter, the square encloses the maximum area.
Considering all possible rectangles with a given perimeter, the square encloses the greatest area.
Proof:
Consider a square of dimensions "x "by "x", the area of which is x^2.
Adjust the dimensions by adding "a" to one side and subtracting "a" from the other side.
This 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.
How can you combine these two statements?
