Lou wrote:

... If you turn up the temperature on the water heater, you'll end up heating
50 gallons of water to the higher temperature when you only need to heat 10.

Heat loss from a hot object to cooler surroundings, in this case from the
water heater itself as well as from the run of pipe from the heater to the
diswasher, is directly proportional to the difference in temperature between
the hot object(s) and the surroundings...

So turning up the heater temperature will likely mean you heat more water
than you need to a higher temperature than you need and increase heat losses.

Turning the tank heater down might save energy, with low-power tankless
electric heaters in the kitchen and bathroom. Boosting 1.25 gpm from T
to 110 F with 5 kW for a shower makes 1.25x8x(110-T)x60 = 3.412x5000, so
T = 82 F. Lots of Caribbean people are happy taking showers at 85 F.

Nick