W = The width of fence
L = Length of house = third side of fence
2 W + L = 60 ft
L = 60 - 2 W
A = W * L
A = W * ( 60 - 2 W )
A = 60 W - 2 W ^ 2
Now you must use calculus :
If a first derivative of a function at some point is equal to zero function has maxsimum or minimum.
If the second derivative is negative then the function has maximum.
If the second derivative f'' is positive then the function has minimum.
d A / d W = 60 - 2 * 2 W = 60 - 4 W
d A / d W = 60 - 4 W = 0
60 = 4 W Divide both sides by 4
15 = W
W = 15 ft
L = 60 - 2 W
L = 60 - 2 * 15
L = 60 - 30
L = 30 ft
Second derivative = - 4
the function has maximum.
Maximum area :
Amax = W * L = 15 * 30 = 450 ft ^ 2
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