Answer:
Step-by-step explanation:
Consider the outer surface of the roof to be

Since, the heat conducted is equal to the sum of the heat transferred through convection and the rest by radiation

Rewrite the equation as follows


subsititute
k = 2 W/m

0.22 for L





![2* (15 -T_(out))/(0.22) =[0.9*(5.67*10^-^8)*((T_(out)+273)^4-255^4)]+[15*(T_(out)-10)]\\\\T_(out)=7.7^oC](https://img.qammunity.org/2021/formulas/engineering/college/9ralzca6qzby3eg9zok0fpicd0gqzwuwor.png)
Hence, the temperature of outer surface of the roof is

Calculate the surface area of the roof

Here, b is the width , l is the length
substitute 15 for b , 20 for l

Write the equation for conduction

substitute 2W/m.K for k


Therefore, the total heat transferred through conduction is

Consider the amount of natural gas required be R and the cost incurred in running the furnace through the night be M

Duration of time T = 14 x 3600s

And the required money for the gas M = 11.2 x $1.2
= $13.44
Therefore, the money lost through the roof due to the heat transfer M=$13.44