Video: Understanding Light-Years

There are 9.46 × 10¹⁵ meters in 1 light-year. How many meters are there in 4 light-years? Give your answer to 2 significant figures.

02:40

Video Transcript

There are 9.46 times 10 to the power of 15 meters in one light-year. How many meters are there in four light-years? Give your answer to two significant figures.

Okay, so, the first sentence of this problem statement is telling us that 9.46 times 10 to the power of 15 meters is the same thing as one light-year. And based on this information, we’re being asked to find how many meters there are in four light-years.

Now to do this, we can take these two equations and divide the left-hand side of the bottom equation by the left-hand side of the top equation. But then, if we do that then it must be equal to the right-hand side of the bottom equation divided by the right-hand side of the top equation.

Now that sounds like a little bit much. So, let’s start by saying that the question mark we put here is actually equal to 𝑥. And 𝑥 is the value that we’re trying to find, the number of meters in four light-years. So, what we’re saying we can do is to divide 𝑥 by 9.46 times 10 to the power of 15 meters, so that’s this quantity on the left-hand side of the top equation. And we set this fraction equal to four light-years divided by one light-year. So, that’s this fraction on the right.

Now the reason that we can do this is because on the left-hand side of this new equation that we’ve created, we’re finding the ratio between this quantity and this quantity. But then, because this quantity is the same thing as this quantity and this quantity is the same thing as this quantity, finding the ratio between 𝑥 and 9.46 times 10 to the power of 15 meters is the same thing as finding the ratio between four light-years and one light-year.

But anyway, so coming back to this equation that we’ve created now, we can rearrange to solve for 𝑥. We can do this by multiplying both sides of the equation by 9.46 times 10 to the power of 15 meters, in other words, the denominator of the fraction on the left. Because this way, the denominator will cancel with this value that we’ve multiplied the fraction by. And so, on the left we’re just left with 𝑥. And on the right, we’ve got four light-years divided by one light-year all multiplied by 9.46 times 10 to the power of 15 meters.

Now we can see on the right-hand side that in our fraction here the unit of light-years in the numerator cancels with the unit of light-years in the denominator. And so, the value of this fraction basically just becomes four divided by one, which is four. And so, the whole thing simplifies to four, meaning that 𝑥 is equal to four multiplied by 9.46 times 10 to the power of 15 meters.

When we evaluate the right-hand side then, we find that 𝑥 is equal to 3.784 times 10 to the power of 16 meters when written in standard form. But, remember, this is not our answer yet. We need to give our answer to two significant figures. And so, we need to round this quantity to two significant figures. So, here’s our first significant figure, and here is our second. Now we look at the next one, which happens to be an eight. And eight is larger than five, so this second significant figure is going to round up. In other words, then, the quantity that we’ve called 𝑥 is equal to 3.8 times 10 to the power of 16 meters to two significant figures.

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