Mercury travels 364 million kilometers as it makes one full orbit around the Sun. Assuming that Mercury has a circular orbit, at what distance away from the Sun does Mercury orbit? Give your answer in scientific notation to two decimal places.
Okay, so this question is about Mercury as it orbits the Sun. And we’re told to assume that this orbit is circular. Let’s suppose that this here is the Sun. And then, we can draw Mercury orbiting around it like this, where the pink circle shows the orbit that Mercury follows. We are told that in order to make one full orbit, Mercury travels a distance of 364 million kilometers. One full orbit means that Mercury travels all the way around the circle in this diagram so that after completing this orbit, it ends up at the same position on this circle that it started at. This means that the distance traveled in one full orbit is equal to the circumference of the circle that Mercury orbits on. We’ll label the circumference of this orbit as 𝐶. So, we have that 𝐶 is equal to 364 million kilometers.
We are asked to work out the distance away from the Sun that Mercury orbits. As we can see from the diagram that we’ve drawn, this distance is the radius of the orbit, and we’ll label it as 𝑟. We can recall that the circumference of a circle is equal to two 𝜋 multiplied by the circle’s radius. In this case, we know the value of the circumference, and we’re trying to work out the radius. So, let’s take this equation and rearrange it to make 𝑟 the subject.
To do this, we divide both sides of the equation by two 𝜋. Then, on the right-hand side, we have a two 𝜋 in the numerator which cancels with the two 𝜋 in the denominator. This leaves us with an equation that says 𝐶 divided by two 𝜋 is equal to 𝑟. And of course, we can also write this the other way around to say that the radius 𝑟 is equal to the circumference 𝐶 divided by two 𝜋. The circumference 𝐶 is equal to 364 million kilometers. Written out as a number, this is 364 followed by six zeros. We can now take this value for 𝐶 and sub it into this equation to calculate the radius 𝑟. Then, we can evaluate this expression to find that 𝑟 is equal to 57932399.29 kilometers, where the ellipses indicate that there are further decimal places.
Now, this is quite a long number. And when we have numbers like this, it’s not generally conventional to write them out in the way that we’ve done here. It’s far more typical to give these numbers in scientific notation. In fact, if we look back at the question, we can see that we are asked to give our answer in scientific notation to two decimal places. In order to write this number in scientific notation, we need to move the decimal point one, two, three, four, five, six, seven places to the left. Then, in scientific notation, we have that 𝑟 is equal to 5.7932 et cetera times 10 to the seven kilometers. The final step is to round this answer to two decimal places. When we do this, we get our answer to the question that the distance away from the Sun that Mercury orbits is equal to 5.79 times 10 to the seven kilometers.