The rectangle will be its largest when the cost of the north/south fencing is equal to the cost of the east/west fencing, and both are half the total.
The north and south dimensions will be ($480/2)/($10/ft)/2 = 12 ft.
The east and west dimensions will be ($480/2)/($15/ft)/2 = 8 ft.
The largest possible garden is 12 ft in the north-south direction by 8 ft in the east-west direction.
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Let p represent the cost per foot in the x direction, and let q represent the cost per foot in the y direction. Then the total cost is
... C = 2px + 2qy
and the area is
... A = xy
In terms of x, the area is
... A = x(C -2px)/(2q)
This describes a downward-opening parabola whose vertex is at x=C/(4p). That is, the value of x will be C/(4p) and the cost in the x-direction will be
.. 2px = 2p(C/(4p)) = C/2
