Virtually any question involving area can be modeled with a quadratic equation.
(1) Find the area of an oil spill of radius r. `A=pir^2` is quadratic in r.
(2) Find the maximum area encompassed by a given length of fence.
If the length of fence is P and the area is a rectangle we have A= l x w and 2l+2w=P so A= l x ((P-2l)/2) is quadratic in l.
(3) We might model the surface area of an animal by a cylinder or sphere. In the case of a cylinder `SA=2pir^2+2pirh` and for a sphere `SA=4pir^2` which are both quadratic in r.
(4) A cable with equally spaced weights hanging from it assumes the shape of a parabola.
(5) Automobile headlights and flashlights have parabolic reflectors -- a parabola in cross-section.
(6) Many receivers -- satellite antennaes, sound receivers, etc... are parabolic.
(7) Some arches are designed as parabolas.