Consider a $1\mathrm{.}5-$m$-$high electric hot$-$water heater that has a diameter of $40$ cm and maintains the hot water at $60C.$

The tank is located in a small room whose average temperature is $30C,$ and the heat transfer coefficients on the

inner and outer surfaces of the heater are $50$ and $12\frac{W}{m^2}*K,$ respectively. The tank is placed in another $46-cm-$

diameter sheet metal tank of negligible thickness, and the space between the two tanks is filled with foam insulation

$).$ The thermal resistances of the water tank and the outer thin sheet metal shell are very small and

can be neglected. The price of electricity is $$\frac{0\mathrm{.}08}{k}Wh,$ and the homeowner pays $$280$ a year for water heating.

Determine the fraction of the hot$-$water energy cost of this household that is due to the heat loss from the tank.

Hot$-$water tank insulation kits consisting of $3-cm-$thick fiberglass insulation $(k=0\mathrm{.}035\frac{W}{m}*K)$ large enough to wrap

the entire tank are available in the market for about $$30.$ If such an insulation is installed on this water tank by the

homeowner himself, how long will it take for this additional insulation to pay for itself?

The fraction of the hot$-$water energy cost of this household that is due to the heat loss from the tank is

The time necessary for this additional insulation to pay for its cost of $$30$ is

months.