Length of an Arc Application

Problem 203: Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day.
Use your answer from part (a) to determine the radian measure for Mercury’s moment in one Earth day.

Solution:

Let’s begin by finding the circumference of Mercury’s orbit.
(1) $\begin{align*} C & = 2 \pi r = 2 \pi (36 \ \text{million miles}) \approx 226 \ \text{million miles}\end{align*}$
Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:
(2) $\begin{equation*} (0.0114)226 \ \text{million miles} = 2.58 \ \text{million miles} \end{equation*}$
Now, we convert to radians:
(3) $\begin{equation*} \text{radian} = \frac{\text{arc length}}{\text{radius}} = \frac{2.58 \ \text{million miles}}{36 \ \text{million miles}} = 0.0717.\end{equation*}$