Question

Consider the following heating of a house. The house has 3,200ft^2 of living space. 80,000 BTUs of heat per square foot are required to heat the house for the winter. Natural gas is available at a cost of $5 per one thousand cubic feet. One cubic foot of natural gas supplies 1,000 BTUs of heat is beefy. The furnace in the house is 70% efficient.

A) calculate the number of cubic feet of natural gas that is required to heat the house for one winter

B) the cost of heating the house for one winter

Answer 1

A) 365714 ft³

B) $1829

Explanation:

You want the volume of gas and the cost of it for supplying 80,000 BTU/ft² to a 3200 ft² house at the rate of 1000 BTU/ft³ and a cost of $5 per 1000 ft³ using a furnace that is 70% efficient.

A) Volume

If the furnaces is 70% efficient, then 70% of the heat energy in the gas goes to heating the house. If V is the volume of gas used, then the heat provided to the house is ...

0.70 × (1000 BTU/ft³) × V = (3200 ft²) × (80,000 BTU/ft²)

V = (3200·80,000)/(0.70·1000) ft³ = 256000/0.7 ft³ ≈ 365714 ft³

The volume of gas required for heat for the winter is about 365714 cubic feet.

B) Cost

The cost of the gas will be ...

365714 ft³ × ($5/(1000 ft³)) = 365.714·($5) ≈ $1829

The cost of heating the house for the winter is about $1829.

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Additional comment

The area of the house is given to 2 significant figures, and some of the remaining numbers are given to 1 significant figure. It seems reasonable to round the answers to 366000 ft³ and $1830. To 1 sf, these become 400,000 ft³ and $2000.

Effectively, about 114.29 ft³ of gas is required for each square foot of floor space, at a cost of $0.57143 per square foot.