The residential rate for natural gas is about $15 per thousand cubic foot. Burning one cubic foot of natural gas releases about 1080 kJ of heat. How much would it cost to heat the water in a 25,000 gallon swimming pool from 52^$\circ$F to 78^$\circ$F, assuming all of the heat from burning the natural gas went towards warming the water (1 gal = 3.785 L; the specific heat of water = 4.184 J/g·^$\circ$C)

Question

Quizplus · Accepted Answer

The first step is to convert the volume of the pool from gallons to liters:
25,000 gal x 3.785 L/gal = 94,625 L
Next, we need to calculate the amount of heat energy required to raise the temperature of the water:
Q = m x c x ΔT
where Q is the heat energy in joules, m is the mass of water in grams, c is the specific heat capacity of water, and ΔT is the change in temperature.

To find the mass of water, we need to know the density of water:
1 g/mL = 1 kg/L = 1000 g/L

So the mass of the water in the pool is:
94,625 L x 1000 g/L = 94,625,000 g

The change in temperature is:
78°F - 52°F = 26°F = 14.44°C

Now we can calculate the amount of heat energy required:
Q = (94,625,000 g) x (4.184 J/g·°C) x (14.44°C) = 5.36 x 10^9 J

To convert joules to kilojoules, we divide by 1000:
5.36 x 10^9 J ÷ 1000 J/kJ = 5.36 x 10^6 kJ

Finally, we need to calculate the cost of the gas required to generate this amount of heat energy:
1 m^3 of natural gas produces 35.3 MJ of heat energy:
5.36 x 10^6 kJ ÷ (35.3 MJ/m^3 x 1000 kJ/MJ) = 152 m^3

At $15 per m^3, the total cost is:
152 m^3 x $15/m^3 = $2,280

Rounding to the nearest dollar gives us the final answer of $79.

The Residential Rate for Natural Gas Is About $15 Per ∘\circ∘

The Residential Rate for Natural Gas Is About $15 Per $\circ$