My family has two showers and whenever someone is using the other shower, my brother raises the temperature setting on the boiler. My brother is trying to argue with me that raising the boiler's temperature setting from 110 degrees Fahrenheit to 125 degrees Fahrenheit will make the water stay hot for a longer period of time. He thinks the physical contact with existing boiler water will make the outlet water hotter therefore water stays hot LONGER.

However, I do not think so. I think as soon as the initial volume of 120-degree water runs out, the water will continue to exit the boiler at the same temperature if the boiler's setting was at 110 degrees. Therefore, he will feel the same temperature water as if the setting was at 110 after some time (when the initial hot water runs out). He showers for 20 minutes which is when it starts to get cold if there are two showers running.

Is his reasoning correct? I thought raising the limiter on the boiler only lets it hold hotter water; I do not think it makes the shower water stay hot LONGER. I did not do well in Thermodynamics, but I've drawn the best representation I can to help visualize the problem. Are there any other properties I need? [![enter image description here](https://i.stack.imgur.com/ha6Gv.jpg)](https://i.stack.imgur.com/ha6Gv.jpg)