length = x
width = y
area = A = x y
length of fencing = 2x+y = 40
y = 40-2x
A = x (40 - 2x)
A = 40 x - 2 x^2
maxmimum A , when dA/dx = 0
0 = 40 - 4 x
x = 10
then y = 40 - 20 = 20
area = A = 200
alternatively look at parabola
2 x^2 -40 x = -A
x^2 - 20 x = -(A/2)
x^2 - 20 x + 100 = -(A/2) + 100
(X-10)^2 = -(1/2)(A+200)
vertex at x = 10, A = 200
dArea/dL=0=40-2L + l(-2)
L=10 W= 20 so the pen is 10x20,with the barn serving as a 20ft side.
graph area vs L. Where is it maximum?
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