Explanation:
x = length
y = width
let's assume the side with the adjoining road is a length. we will see that given the properties of a rectangle it does not matter. we will get the same result either way.
the perimeter is
2x + 2y = perimeter
the area is
x × y = area
we know that
2×8×y + 1×8×x + 1×12×x = 1200
16y + 20x <= 1200 (but can safely take this as "=" to get the maximum area based on the maximum possible perimeter)
20x = 1200 - 16y
x = (1200 - 16y)/20 = 60 - 16y/20 = 60 - 4y/5
we use this in the area equation :
area = f(y) = (60 - 4y/5) × y = 60y - 4y²/5
this function has an extreme value at the zero of the first derivative :
f'(y) = 60 - 8y/5 = 0
-8y/5 = -60
8y = 300
y = 300/8 = 75/2 = 37.5 ft
x = 60 - 4y/5 = 60 - 4×37.5/5 = 30 ft
the second derivative tells us, if it is a max. or a min.
if it is positive, the point is a relative minimum, and if it is negative, the point is a relative maximum.
f''(y) = -8/5 negative for y = 37.5, so it is a max.
now, if we assume that the side with the adjoining road is a width, we only need to switch x and y for each other in all the equations above. we get f(x) instead of f(y), but that does not matter, we get the same equations with the same results : one type of side is 30 ft, the other is 37.5 ft long. the side with the adjoining road is of the 30 ft type.
perimeter = 2×30 + 2×37.5 = 60 + 75 = 135 ft
area = 30 × 37.5 = 1,125 ft²
the dimensions that form the maximum area are
30 ft × 37.5 ft.
one of the 30 ft sides is along the adjoining road.
this max. area is therefore 1,125 ft²