Final answer:
The energy stored in a 50-ampere-minute 12-volt battery is 36,000 joules. To charge the battery using a hydroelectric dam at 100% efficiency, 0.073 cubic meters of water behind a 50 cm high dam would be needed, making 0.1 cubic meters (B) the closest answer.
Step-by-step explanation:
To estimate the energy stored in a 50-ampere-minute 12-volt battery, we calculate the energy using the formula for electrical potential energy, which is Energy (E) = Voltage (V) × Charge (Q). Since 1 ampere-minute is equal to 60 coulombs (C), a 50-ampere-minute charge is 3000 C (50 × 60). Therefore, the energy stored is E = 12 V × 3000 C, which equals 36,000 joules (J).
Next, to determine how much water would be required to generate this amount of energy with 100% efficiency behind a hydroelectric dam of height 50 cm, we use the potential energy formula for gravity, PE = m × g × h, where m is the mass of water, g is the acceleration due to gravity (9.81 m/s²), and h is the height. Since the energy is 36,000 J, and water has a density of 1000 kg/m³ (or 1 kg/L), we can find the volume of water needed by rearranging the formula to V = E / (g × h × density). Plugging in the values, we get V = 36,000 J / (9.81 m/s² × 0.5 m × 1000 kg/m³), which is approximately 0.073 cubic meters of water.
The closest answer to 0.073 cubic meters from the given options is 0.1 cubic meters (B).