My hot water heater is, like, 15 feet from my shower
1/2 copper pipes seem pretty small
And yet, based on what I estimate is the flow rate of my shower, there’s about 100 gallons of cold water sitting in those hot water pipes that I need to purge while I stand.
Because it’s California and I’m trying to be good and save water, I use a bucket to collect the cold water that runs until I get hot enough water to shower. My heater is about 30 feet from the shower. I get three gallons before the water is shower temperature. I’d estimate it’s less than half that before the pipe is purged. I think the rest goes to gradually warming the pipes. 100 gallons sounds high by a couple of orders of magnitude.
We can disregard that number as hyperbolic exaggeration. A high capacity water heater is 75 gallons, which would mean that flushing 100 gallons of cold water from the pipes literally drains every drop of hot water from the water heater plus 25 further gallons from the heater’s inlet.