### Video Transcript

Jupiter takes 4333 days to orbit the Sun. How long is this period in Earth years? Use a value of 365 for the number of days in an Earth year. Give your answer to one decimal place.

Okay, so this first part of the question is asking us how long a time of 4333 days is in units of Earth years. We are told to use a value of 365 for the number of days per Earth year. In order to make the distinction clear between Earth years and years on Jupiter, we will label units of Earth years as 𝑦 subscript E. Since there are 365 days per Earth year, then we can equivalently write this to say that there’s one over 365 Earth years per day. In other words, each one of the 365 days in an Earth year makes up one three hundred and sixty-fifth of that year. Now, just as we can write the units meters per second as units of meters divided by units of seconds, we can also write Earth years per day as the units of Earth years divided by the units of days.

In order to convert our time of 4333 days into units of Earth years, we need to multiply our time in units of days by one over 365 Earth years per day. If we look at the units in this expression, we see that the units of days cancel with the units of per day. And so we are left with units of Earth years. Then, all that’s left to do is to multiply 4333 by one over 365. When we do this, we get a result of 11.871 Earth years, where the ellipses indicate that there are further decimal places. Looking back at the question, we see that we were asked to give our answer to one decimal place. Rounding to one decimal place gives a result of 11.9 Earth years. So our answer to this first part of the question is that 4333 days in Earth years is equal to 11.9 years. This result means that it takes Jupiter 11.9 Earth years to orbit the Sun.

Now, let’s look at the second part of the question.

How long does Earth take to orbit the Sun in Jupiter years? Give your answer to two decimal places.

Now, when we answered the first part of this question, we found that a time of 4333 days was equal to 11.9 Earth years. Since the question tells us that Jupiter takes this time of 4333 days to orbit the Sun, then that means that 4333 days is equal to one year on Jupiter. We have labeled a Jupiter year as 𝑦 subscript J. Since this time of 4333 days is equal to 11.9 Earth years and is also equal to one Jupiter year, then we know that 11.9 Earth years must be equal to one Jupiter year. We are being asked to work out how long Earth takes to orbit the Sun in units of Jupiter years. By definition, the Earth orbits the Sun in a time of one Earth year. So what we’re trying to do is to find out how many Jupiter years is equal to one Earth year.

If we take this equation and divide both sides by 11.9, then on the left-hand side, we have 11.9 divided by 11.9, which gives a result of one. And so we have that one Earth year is equal to one over 11.9 in units of Jupiter years. Evaluating one over 11.9 gives a result of 0.084, where the ellipses indicate that there are further decimal places. So we have that one year on Earth is equal to 0.084 years on Jupiter. In other words, Earth takes 0.084 Jupiter years to orbit the Sun. The question asks us to give our answer to two decimal places. Rounding to two decimal places gives us that one Earth year is equal to 0.08 Jupiter years. And so our answer to the second part of the question is that in units of Jupiter years, it takes Earth 0.08 years to orbit the Sun.

Now, for the final part of the question.

One day on Jupiter is 9.925 hours long. How many Jupiter days are in one Earth year? Use a value of 365 for the number of days in an Earth year. Round your answer to the nearest day.

Okay, so this last part of the question is asking us how long a time of one Earth year is in units of Jupiter days. We are told that one day on Jupiter is 9.925 hours long. We’ll denote units of Jupiter days as 𝑑 subscript J. We also know that one Earth day, which we’ve labeled 𝑑 subscript E, is equal to 24 hours, since each day on Earth is 24 hours long. We’re told in the question to use a value of 365 for the number of days in an Earth year. So we can say that one Earth year is equal to 365 Earth days. We are being asked to work out how many Jupiter days are in one Earth year. This means we need to find the result of converting one Earth year into units of Jupiter days.

We know that one Earth year is equal to 365 Earth days. And we also know that each one of those 365 Earth days is equal to 24 hours. If one Earth day is equal to 24 hours, this means that there are 24 hours per Earth day. So to convert our expression for one Earth year in units of Earth days into units of hours, we need to multiply by 24 hours per Earth day. Then, in terms of the units, the Earth days cancel with the per Earth day, and we are left with units of hours. Doing the multiplication gives us that one year on Earth is equal to 8760 hours.

So we now know what one Earth year is in units of hours. And we also have an equation that relates units of hours to units of Jupiter days. Specifically, we know that one Jupiter day is equal to 9.925 hours. So we can say that there are 9.925 hours per Jupiter day, or, equivalently, we can say that there is one over 9.925 Jupiter days per hour. Then, to convert our expression for one Earth year in units of hours into units of Jupiter days, we need to multiply by one over 9.925 Jupiter days per hour. The units of hours cancel with the units of per hour, and we are left with units of Jupiter days.

Evaluating this expression gives us that one Earth year is equal to 882.6196 Jupiter days, where the ellipses indicate that there are further decimal places. The question asks us to round our answer to the nearest day. So we need to take this value here and round it to the nearest whole number of Jupiter days. When we do this, our result rounds up to give 883 Jupiter days. And so we have our answer to the final part of the question that the number of Jupiter days in one Earth year is 883 days.