Your numbers are somewhat different from mine. I did my math for a Cherokee, but that shouldn't result in such a difference. All numbers at max gross, ISA conditions.
You need about 1600 ft of runway to clear a 50ft obstacle.
If dealing with a short runway, you climb at Vx, not Vy. So 75mph, climb at 800-1000 fpm, takes 35 seconds or so to gain 500 ft. You cover about 4000 ft horizontally in the process.
At that point the engine quits.
You initiate a best glide (85mph) 45° bank angle turn back to the runway.
Turn radius will be approx 500 ft in those conditions. It will take about 15 seconds to turn back approx 210°. In those 15 seconds you lost about 210 feet.
You are 5600 ft away from the start point of your take-off. You'll cover another 2600ft with your remaining altitude, not taking into account that you'll float in ground effect and you have some extra energy as you flare and bleed speed.
So, in theory, if the runway is 3000ft long, you'll (barely) make it back from 500ft AGL.
Apologies, I meant to say Cessna 182, not 172. However, I don't believe the difference is significant. I've compared my calculations with results from both MS Flight Sim and Redbird, and the agreement was very good. Additionally, please note that I’m assuming a "normal" takeoff, not a best-performance takeoff with an aggressive climb angle as one might do if they knew ahead of time that the engine was going to fail. If you do a best performance takeoff, then my calculations do agree with your numbers somewhat (see details below):
Here are my assumptions:
Weight = 3100lb (max gross)
Ground roll = 2000 ft (best performance is 800 ft, but I am assuming a longer roll to account for the fact that you can't transition from liftoff to 1000 fpm climb instantly. The extra time translates to extra distance.)
Climb speed = 85 knots (best performance is 60 knots, but I am assuming a more leisurely 85 knots)
Climb rate = 1000 fpm (this is close to the published best performance rate)
Reaction time after engine failure = 3 seconds (this is the delay time between engine failure and start of the turn. This is an optimistic number because I am assuming the bank angle is achieved instantly, but in reality it takes a few extra seconds to establish the bank)
Glide speed = 80 knots (POH also recommend 80 knots)
L/D ratio = 11.0 (this is taken from the published best glide angle)
Bank angle = 40-deg (40-50deg seems to be the sweet spot)
Wind = 0
Temperature = ISA
Runway length = 3500 ft
Calculated values:
Climb angle = 6.6 deg
Descent angle = 5.2 deg
Altitude at Departure End of Runway (DER) = 175 ft
Descent rate during wings level = 740 fpm
Descent rate during turn = 960 fpm
Turn radius = 880 ft
Assuming engine fails at 1350 ft:
Distance from DER at engine failure = 10,100 ft
Altitude after 3 second reaction time = 1300 ft
Distance from runway after reaction time = 10,500 ft
Turn required = 190-deg
Altitude at end of turn = 960 ft
Diagonal distance from runway at end of turn = 10,500 ft
Glide time to runway = 78 sec
Glide time available = 78 sec (just barely make the runway)
Assuming engine fails at 600 ft:
Distance from DER at engine failure = 7150 ft
Altitude after 3 second reaction time = 560 ft
Distance from runway after reaction time = 7600 ft
Turn required = 205-deg
Altitude at end of turn = 190 ft
Diagonal distance from runway at end of turn = 4100 ft
Glide time to runway = 30 sec
Glide time available = 15 sec (cannot make the runway)
Assuming a more aggressive climb - same assumptions as before, except:
Ground roll = 1000 ft
Climb speed = 60 knots
Bank angle = 45-deg
Calculated values:
Climb angle = 9.3 deg
Descent angle = 5.2 deg
Altitude at Departure End of Runway (DER) = 410 ft
Descent rate during wings level = 740 fpm
Descent rate during turn = 1150 fpm
Turn radius = 740 ft
Assuming engine fails at 600 ft:
Distance from DER at engine failure = 4650 ft
Altitude after 3 second reaction time = 560 ft
Distance from runway after reaction time = 4950 ft
Turn required = 235-deg
Altitude at end of turn = 130 ft
Diagonal distance from runway at end of turn = 1450 ft
Glide time to runway = 11 sec
Glide time available = 11 sec (just barely make the runway)