### Video Transcript

A manufacturer makes a type of
solar panel that has a power output of 290 watts when exposed to direct
sunlight. Eight of these solar panels are
fitted to the roof of a house, which receives an average of 3.2 hours of direct
sunlight per day over the course of a year. What is the total energy output of
the solar panels on this house over the course of a year? Assume that there are 365 days in a
year. Give your answer in kilowatt hours
to the nearest kilowatt hour.

Okay, so this is a really long
question, which means we need to underline all the important bits so we don’t miss
anything out. So first things first, we’re told
that a manufacturer makes a type of solar panel that has a power output of 290
watts. And this power output is when the
solar panel is exposed to direct sunlight. Now we’re also told that eight of
these solar panels are fitted to the roof of a house. And the roof of this house receives
on average 3.2 hours of sunlight — direct sunlight — per day and this is over the
course of a year.

What we’ve been asked to do is to
find the total energy output of the solar panels on the house over the course of a
year. We’ve also been told to assume that
there are 365 days in a year and we have to give our answer in kilowatt hours to the
nearest kilowatt hour. So let’s label some of the
quantities that we’ve been given in the question.

First of all, we know the power
output of a solar panel. This power output is 290 watts,
which we’ll call 𝑃. Now, it’s important to note that
this power output only occurs when the solar panel is exposed to direct
sunlight. Luckily for us, however, in the
question, we’re told that the eight solar panels that are fitted on the house
receive on average 3.2 hours of direct sunlight per day. Therefore, we’ve only been told how
much direct sunlight the panels receive. And so we don’t need to worry about
the fact that direct sunlight has been mentioned. Anyway, so eight of these panels
are fitted onto a roof.

And we’ve also been told that the
roof itself receives an average of 3.2 hours of direct sunlight every day and this
is over the course of a year. In other words, every day on
average, there’re 3.2 hours of sunlight and this has been measured over a year. Now of course, some days will be
more sunny and some days will be less sunny. But on average, over a year, we
find that there are 3.2 hours of direct sunlight per day on the roof.

Now what we’ve been asked to find
is the total energy output by the solar panels — that eight solar panels on the roof
— over the course of one year. We’ll call this 𝐸 sub year. And we need to give our answer in
kilowatt hours to the nearest kilowatt hour. The last piece of information that
we know is that we have to assume that there are 365 days in a year. Now, we might think “of course!
there are 365 days in a year.” Well, not really, first of all, it
could have been a leap year, which means that there is an extra day in the year. So there are 366 days.

But also in reality, the number of
days in a year are slightly more than 365. It’s about 365 and a quarter days
per year. That’s why we have leap years in
the first place. We just say that three years have
365 days each and then the fourth year has 366 days to acount for that extra
one-quarter days for each of the past four years. But anyway, so we’ve been told to
assume that there are 365 days in a year.

Okay, so here’s one of our solar
panels. And this converts light energy into
290 watts of power. That’s for one panel. But we know that on top of this
roof that we’re studying, we’ve got eight of these panels. Therefore, the total power output
which we’ll call 𝑃 sub tot is equal to 290 times eight because each panel produces
290 watts of power and we’ve got eight of these panels on the roof. Now, this happens to be 2320
watts.

However, since we need to give our
final answer in kilowatt-hours, it’s best if we convert this power into
kilowatts. We can do this by recalling that
one kilowatt is equal to 1000 watts. But here, we have 2320 watts or in
other words we have 2.32 kilowatts because we have 2.32 lots of 1000 watts and
that’s the same as 2320 watts. And hence, 𝑃 sub tot the total
power output of the eight solar panels combined is 2.32 kilowatts. So that’s the power of the
panels. But we need to work out how much
energy they’re putting out over the course of a year.

To do this, we can recall that
power is defined as the energy per unit time. More specifically, it’s the energy
transferred per unit time. Now, in the case of these solar
panels, that’s the energy output by the solar panels divided by the time taken for
this energy to be output. Now, we can use this equation to
work out the amount of energy output by the panels every day.

To do this, we first multiply both
sides of the equation by the time 𝑡. This way the time cancels on the
right-hand side, leaving us with the time multiplied by the power is equal to the
energy. Now, 𝑡 refers to the number of
hours of sunlight we have every day and 𝑃 is the power output of the solar
panels. Therefore, 𝐸 is the amount of
energy output by the solar panels every day. So let’s call this energy 𝐸 sub
day because that’s the amount of energy output every day.

So the amount of time for which the
sun is shining on the solar panels every day is 3.2 hours as we’ve been told. And as we’ve worked out the total
power of all the eight solar panels combined is 2.32 kilowatts. And hence, the product of these two
quantities is going to give us the amount of energy output by the solar panels every
day.

And we can look at the units very
quickly. We’ve got hours and we’ve got
kilowatts. So the unit of the final quantity
when we multiply these two together is going to be hours times kilowatts or kilowatt
hours. And this is what we want our final
answer in. So we are working on the right
lines anyway so the left-hand side evaluates to 7.424 kilowatt-hours. That’s the amount of energy we
output from the solar panels every day.

So how much energy do we output
every year? Well, if this is the energy output
every day and we know that there are 365 days in one year, then we can say that the
energy output per year is equal to 365 — that’s the number of days in every year —
times the energy output per day. And happily for us, we already know
what 𝐸 sub day is. So we can substitute in 7.424
kilowatt-hours. When we evaluate the product, we
find that the amount of energy output by the solar panels every year is 2709.76
kilowatt-hours.

But we cannot stop here because
remember we need to give our answer to the nearest kilowatt-hour. In other words, we need to round
this digit here — the one just before the decimal. Now, it’s the one after the decimal
that will tell us whether nine rounds up or stays the same. Well, this value here is a seven
and that’s larger than a five. Therefore, this nine is going to
round up to a 10. Well, yes, it’s going to become a
10. So we’ve got the zero here and
there’s a one. But the one can carry forward into
the tens column. And therefore, 2709.76 runs up to
2710 to the nearest whole number.

And at this point, we have our
final answer. The total energy output of the
solar panels on the house over the course of one year is 2710 kilowatt-hours to the
nearest kilowatt-hour.