A farmer has 1000 yds of fencing to enclose a rectangular garden. Express the area A as...?

a) express the area A as a function of the width x of the rectangle.


b) what is the 'real world' domain of A?


c) what is the vertext of A(x)?


d) what is the maximum area the farmer can enclose?


please show work (the steps taken)

A farmer has 1000 yds of fencing to enclose a rectangular garden. Express the area A as...?
Part a)





x = width


y = length


2x + 2y = 1000, so y = 500 - x





A = xy = x(500 - x) = 500x - x^2





b) x should be positive (and small enough to fit the plot of land that he is planting the garden in)





c) If one corner is at the coordinates (0,0), the opposite corner is at (x,y) as defined above.





d) A = 500x - x^2





Calculus Solution:





dA/dx = 500 - 2x


When 500 - 2x = 0, the area is a maximum. Here, x = 250 and y = 250, so the maximum area is 62,500 square yards.





Non-calculus Solution:





Graph the equation A = 500x - x^2. You can see that the parabola crosses the x-axis at x = 0 and at x = 500. By symmetry, you know that the peak of the parabola must be half-way between 0 and 500 at x = 250. When x = 250, A = 62,500, which is the maximum area.
Reply:Assuming the area to be a square, the square root will give the side. Then calculate area.