For his 18th birthday in February Peter plants to turn a hut in the garden of his parents into a swimming pool with an artifical beach. In order to estimate the consts for heating the water and the house , peter obtains the data for the natural gas combustion and its price.
Calculate the energy (in MJ) which is needed to maintain the temperature inside the house at `30.0^(@)C` during the party (12 hours) .
`1.00 m^(3)` of natural gas as delivered by PUC costs 0.40 € and `1.00 k` Wh of electricity costs 0.137 € . The rent for the equipment for gas heating will cost him about 150.00 € while the corresponding electrical heaters will only cost 100.00 €.

Answer

To solve the problem of calculating the energy needed to maintain the temperature inside Peter's house at 30.0°C during his birthday party for 12 hours, we will break it down into two parts: the energy required to heat the water in the swimming pool and the energy required to heat the air in the house.

### Step 1: Calculate the energy required to heat the water

1. **Determine the volume of water**:
- Given volume of water = 22.5 m³.

2. **Convert volume to mass**:
- Density of water = 1000 kg/m³.
- Mass of water (m) = Volume × Density = 22.5 m³ × 1000 kg/m³ = 22,500 kg.

3. **Calculate the energy required to heat the water**:
- Specific heat capacity of water (Cₚ) = 4.18 kJ/kg·°C.
- Assume the initial temperature of water is 16°C (for example).
- Change in temperature (ΔT) = Final temperature - Initial temperature = 30°C - 16°C = 14°C.
- Energy required (E_water) = m × Cₚ × ΔT = 22,500 kg × 4.18 kJ/kg·°C × 14°C.
- E_water = 22,500 × 4.18 × 14 = 1,323,060 kJ.

### Step 2: Calculate the energy required to heat the air

1. **Determine the volume of the house**:
- Assume dimensions of the house are given as 15 m × 8 m × 3 m.
- Volume of the house = 15 m × 8 m × 3 m = 360 m³.

2. **Convert volume to mass**:
- Density of air ≈ 1.225 kg/m³.
- Mass of air (m_air) = Volume × Density = 360 m³ × 1.225 kg/m³ = 441 kg.

3. **Calculate the energy required to heat the air**:
- Specific heat capacity of air (Cₚ) = 1.005 kJ/kg·°C.
- Change in temperature (ΔT) = 30°C - 16°C = 14°C.
- Energy required (E_air) = m_air × Cₚ × ΔT = 441 kg × 1.005 kJ/kg·°C × 14°C.
- E_air = 441 × 1.005 × 14 = 6,141.87 kJ.

### Step 3: Total energy required

1. **Total energy (E_total)**:
- E_total = E_water + E_air = 1,323,060 kJ + 6,141.87 kJ = 1,329,201.87 kJ.

### Step 4: Convert to MJ
1. **Convert kJ to MJ**:
- 1 MJ = 1000 kJ.
- E_total in MJ = 1,329,201.87 kJ / 1000 = 1,329.20 MJ.

### Conclusion
The total energy needed to maintain the temperature inside the house at 30.0°C during the party for 12 hours is approximately **1,329.20 MJ**.

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