Math / Data & Statistics

QuestionWhen it rains, water vapor in the air condenses into liquid water, and energy is released. (a) How much energy is released when 0.0479 m ( 1.89 inch) of rain falls over an area of $2.59 \times 10^{6} \mathrm{~m}^{2}$ (one square mile)? (b) If the average energy needed to heat one home for a year is $1.50 \times 10^{11} \mathrm{~J}$ , how many homes could be heated for a year with the energy

Studdy Solution

STEP 1

1. The density of water is $1000 \, \text{kg/m}^3$ .
2. The latent heat of vaporization of water is $2.26 \times 10^6 \, \text{J/kg}$ .
3. The volume of rain is calculated as the product of rainfall depth and area.
4. Energy released is calculated using the mass of water and latent heat of vaporization.
5. The energy needed to heat one home for a year is $1.50 \times 10^{11} \, \text{J}$ .

STEP 2

1. Calculate the volume of rain.
2. Calculate the mass of rain.
3. Calculate the energy released.
4. Calculate the number of homes that can be heated.

STEP 3

Calculate the volume of rain using the formula:
$\text{Volume} = \text{Rainfall depth} \times \text{Area}$
Given: - Rainfall depth = $0.0479 \, \text{m}$ - Area = $2.59 \times 10^6 \, \text{m}^2$
$\text{Volume} = 0.0479 \, \text{m} \times 2.59 \times 10^6 \, \text{m}^2 = 1.24161 \times 10^5 \, \text{m}^3$

STEP 4

Calculate the mass of rain using the formula:
$\text{Mass} = \text{Volume} \times \text{Density}$
$\text{Mass} = 1.24161 \times 10^5 \, \text{m}^3 \times 1000 \, \text{kg/m}^3 = 1.24161 \times 10^8 \, \text{kg}$

STEP 5

Calculate the energy released using the formula:
$\text{Energy} = \text{Mass} \times \text{Latent heat of vaporization}$
$\text{Energy} = 1.24161 \times 10^8 \, \text{kg} \times 2.26 \times 10^6 \, \text{J/kg} = 2.8070336 \times 10^{14} \, \text{J}$

STEP 6

Calculate the number of homes that can be heated using the formula:
$\text{Number of homes} = \frac{\text{Total energy}}{\text{Energy per home}}$
$\text{Number of homes} = \frac{2.8070336 \times 10^{14} \, \text{J}}{1.50 \times 10^{11} \, \text{J/home}} \approx 1871 \, \text{homes}$
The energy released can heat approximately $1871$ homes for a year.

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