Question 888586
L = length
W = width
 

Perimeter is normally equal to 2L + 2W, but you are missing 1 side.
We'll assume it's the length.


since you have 1000 feet of fencing, then L + 2W = 1000


solve for L to get:


L = 1000 - 2W


area is equal to L*W


replace L with 1000 - 2W to get:


area = (1000 - 2W) * W


simplify this to get:


area = 1000W - 2W^2


this is a quadratic equation, so convert it to standard form to get:


area = -2W^2 + 1000W


replace W with x to get:


area = -2x^2 + 1000x


since it's in standard form of y = ax^2 + bx + c:


a = -2
b = 1000
c = 0


since the coefficient of the x^2 term is negative, this will have a maximum.


the maximum point will be at x = -b/(2a) = -1000 / -4 = 250.


the equation will be at a maximum when x = 250


when x = 250, -2x^2 + 1000x = -2(250)^2 + 1000(250) = 125000.


since x represents W which represents the width, then W = 250.


since area is L * W, then L * 250 = 125000.


solve for L to get L = 125000 / 250 = 500


the length is 500 and the width is 250.


the area is 500 * 250 = 125000


the perimeter for fencing is L + 2W = 500 + 2(250) = 500 + 500 = 1000


all is good.