I slightly regret that I no longer teach as this would make a brilliant real life maths lesson
ok.
so it seems to be the case that the standard uk bath takes 300 litres if full, less if not.
freestanding and claw baths take more.
info from here:
https://royalbathrooms.co.uk/blog/how-much-water-does-a-bathtub-hold
so lets run two scenarios, a highest possible cost scenario and a lowest likely.
highest: it takes 300 litres
lower: not actually totally full and say 250
cost of water - if the house is not on a meter and is just paying a regular ant per month it’s basically zero extra cost for the baths.
on a water meter - my water company charge 2.95 per cubic metre which is 1000 litres.
https://www.anglianwater.co.uk/services/extra-support/tariff-options/our-standard-charges
So that’s 88.5p for the water.
then there’s heating it up.
again, a lot depends on how it is heated.
I used to live in a very old house and we ran the Aga all the time b the winter so we had hot water on demand because it was generated as a side effect of heating the house.
most newer houses don’t have that so it’s either gas or electric.
i’ll be honest I’m not sure my physics is up to this but for what it’s worth I did measure how much it cost to heat my hot tub up from 18 degrees to 38 degrees and it was ten quid (and that was in January and includes household electricity use overnight)
so as my hot tub is 1800 litres and the bath is 300 litres then 300 to 1800 is one sixth so that’s one pound sixty to heat the water.
gas would be a lot cheaper.
so - lowest likely is no charge for the extra water plus maybe a quid to heat it
highest is 88.5 for the water and 1.60 to heat it is 2 pound 48 and a half p.