Over the course of eight hours, a solar panel in direct sunlight produces a total energy output of 7.92 megajoules. What is the average power output of the solar panel? Give your answer to three significant figures.
Okay, So in this question, we’re dealing with a solar panel that we’ve been told has been kept in direct sunlight for eight hours. And so we can say that the amount of, time we’ll call this 𝑡, for which the solar panel is in direct sunlight is eight hours. And over this period of time, the total energy output of the solar panel — we’ll call this 𝐸 — is 7.92 megajoules. Now, we’ve been asked to find the solar panels average power output. We’ll call this 𝑃. And to work this out, we need to recall the relationship between power, energy, and time. We can recall that power is defined as energy transferred per unit time. And in the case of the solar panel, this is the power produced by the solar panel. This is the energy produced by the solar panel. And this is the time taken for the energy to be produced.
Now, the reason we’re being asked to find the average power output is because we only know the total energy produced over the entire span of eight hours. We don’t know whether that energy was produced at a constant rate over those eight hours or whether at sometimes during those eight hours, the solar panel was producing more energy per unit time and at other times it was producing less energy per unit time. But the point is that we know over those entire eight hours 7.92 megajoules of energy were produced. And so if we calculate the total energy produced divided by the total time taken, then we will find the average power output of the solar panel, which means that we’ll have to substitute these values here into our equation to calculate the average power output. But before we can do this, we need to convert both quantities, 𝑡 and 𝐸, into base units.
Let’s recall that the base unit of time is the second. And so we need to convert this time of eight hours into seconds. To do this, we can remember that there are 60 minutes in every hour. And so we can take our equation 𝑡 is equal to eight hours and multiply the right hand side by 60 minutes — we’ll write that a 60 min — divided by one hour. Now, the reason we can do this is because 60 minutes is exactly the same thing as one hour. And so essentially, this fraction is equivalent to one hour divided by one hour. And anything divided by itself is just one. And so multiplying by this whole fraction means that we’re just multiplying eight hours by one, which is still eight hours.
But the benefit of multiplying by this friction is that now we’ve got a unit of ours in the numerator and the denominator which cancel leaving us with the time 𝑡 is equal to eight multiplied by 60 minutes. And that ends up being 480 minutes. Now, we can do a similar conversion between minutes and seconds because we can recall that there are 60 seconds in every one minute. And so once again, we’re just multiplying our quantity by one. But the unit of minutes in the numerator and the denominator cancel leaving us with the time 𝑡 is equal to 480 multiplied by 60 seconds. This ends up being 28800 seconds, at which point we’ve converted our time 𝑡 from eight hours into seconds, which happens to be 28800 seconds. So now we can move on to looking at the energy.
We can recall that the base unit of energy is the joule. And luckily for us, we’ve been given this quantity in megajoules. So all we need to do is to recall that the prefix mega just means 10 to the power of six. And so an energy of 7.92 megajoules is the same thing as 7.92 times 10 to the power of six joules, at which point we’ve written our energy in the base unit of joules. And so we’ve got both the quantities we need to plug into our equation in their base units now. Which means that we can say that the average power produced by the solar panel is equal to the total energy produced by the solar panel. Which we’ve seen is 7.92 times 10 to the power of six joules divided by the time taken for this energy to be produced, which is 28800 seconds.
Now, very quickly, we can see that we’re going to have a numerical value of 7.92 times 10 to the power of six divided by 28800. And the unit is going to be joules divided by seconds or joules per second, which happens to be the base unit of power, also known as the watt. In other words, then our answer is going to be 7.92 times 10 to the power of six divided by 28800 watts, which ends up being 275 watts. And this value is to one, two, three significant figures, which is exactly what we’ve been asked to give our answer to in the question. So don’t be fooled. If the question asks you to give an answer to say three significant figures. That doesn’t necessarily mean that we’re going to have to do some rounding upon finding our answer. In this case, our answer already happened to be written to three significant figures. And hence our final answer is that the average power output of the solar panel is 275 watts.