I have an electric hot water tank. Recently I've been noticing that the supply of hot water has gone down drastically. It will only be really hot for a couple minutes, warm for another 10 minutes, then cold. I thought I'd check the temperature settings and when I opened the panels on the tank, I noticed that the inside of the upper panel is nice and clean, but the lower panel is very dirty with some kind of residue or corrosion. The insulation looks like it's been scorched, and the thermostat and element access are covered with a kind of blackish residue. This leads me to think that maybe the lower element has failed.
The element is probably fried; it can be checked with an ohm meter with power off. Depending on the wattage of the element it should be ~12-50 ohms as a ball park A dead short or very high reading is bad.
Your lower element is very likely dead. It also looks as if you had a short, possibly caused by a leak . Changing these elements is simple and there are plenty of videos on Youtube.