The output power of the Sun is 1.0 times 1026 W. Calculate, at 8 cents per kWh, the cost of running the sun for 1 hour.
The correct answer and explanation is :
To calculate the cost of running the Sun for 1 hour, given its output power and the cost per kilowatt-hour (kWh), let’s break down the steps.
Step 1: Understand the problem
- Power output of the Sun: ( P = 1.0 \times 10^{26} ) W (watts).
- Cost per kWh: 8 cents per kWh = $0.08 per kWh.
- Time duration: 1 hour.
Step 2: Convert the Sun’s power to kilowatts
1 watt (W) is equal to ( 1/1000 ) kilowatts (kW), so we first need to convert the Sun’s power from watts to kilowatts.
[
P_{\text{kW}} = 1.0 \times 10^{26} \, \text{W} \times \frac{1 \, \text{kW}}{1000 \, \text{W}} = 1.0 \times 10^{23} \, \text{kW}
]
Step 3: Calculate energy produced in 1 hour
Energy is power multiplied by time. Since 1 hour is 1 hour, or 3600 seconds, we can compute the energy output of the Sun in 1 hour:
[
E = P_{\text{kW}} \times \text{time} = 1.0 \times 10^{23} \, \text{kW} \times 1 \, \text{hour} = 1.0 \times 10^{23} \, \text{kWh}
]
Step 4: Calculate the cost
Now, to find the cost of running the Sun for 1 hour, we multiply the energy output by the cost per kWh:
[
\text{Cost} = E \times \text{Cost per kWh} = 1.0 \times 10^{23} \, \text{kWh} \times 0.08 \, \text{dollars/kWh}
]
[
\text{Cost} = 8.0 \times 10^{21} \, \text{dollars}
]
Final Answer:
The cost of running the Sun for 1 hour at 8 cents per kWh is:
[
8.0 \times 10^{21} \, \text{dollars}.
]
Explanation:
The calculation involves understanding the immense scale of the Sun’s energy output. The Sun releases an astronomical amount of energy every second. By converting that to kilowatt-hours and multiplying by the given cost per unit of energy (kWh), we arrive at an enormous figure that highlights the sheer scale of energy the Sun generates. This number, ( 8.0 \times 10^{21} ) dollars, is so large it underscores just how much energy is being radiated by the Sun, far beyond the scope of any human-run energy system. The comparison gives an appreciation for the vastness of the Sun’s power, which is far beyond what we use on Earth in energy consumption.