We know that the calorie is a unit of energy defined as the amount of energy needed to raise 1 gram of water by 1 deg. Celsius.
a) How many calories are required to bring a pot of water at 1 deg. C to a boil? The pot is full to the brim, with a diameter 20cm and depth 20cm. The density of water is 1000 kg/m3?
b) If we consider the characteristic dimension D for the pot is 20cm, approx. how much more (scaling law) is needed to heat a hot tub with D = 2 m? How many calories is that?
c) If energy costs $0.10 /kW-h, how much does it cost to heat this hot tub?
d) How does the price (in dollars) of heating the hot tub scale with the tub's size?
e) What percentage of the cost of heating the hot tub would be saved by reducing the tub's characteristic dimension by 33%?