A coin has the image of \(\pi\) on one side and is being rolled along a horizontal surface. It starts at position \(A\), with the \(\pi\) symbol oriented correctly. The coin then rolls until it reaches position \(B\), which is \(9\) cm to the right of \(A\). If the image of \(\pi\) is now rotated \(90\degree\) counterclockwise for the first time, then what is the radius of the coin? Round your answer to \(1\) decimal. What orientation will the image of \(\pi\) be in after it has rolled a total of \(174\) cm?
At position \(B\), the coin has rotated \(90\degree\) counterclockwise for the first time. Thus, it must have gone through \(\frac{3}{4}\) of a full rotation. Since the distance from \(A\) to \(B\) is \(9\) cm and the coin has gone through \(\frac{3}{4}\) of a full rotation, a full rotation would move the coin \(12\) cm along the line from position \(A\).
A full rotation is also completed by travelling one measure of the circumference of the coin.
Therefore, the circumference of the coin is equal to \(12\) cm.
We also know that \(C=2\times\pi\times r\), where \(C\) is the circumference of a circle and \(r\) is its radius.
Thus, \(12 = 2 \times \pi \times r\). Solving, we have \(r = \dfrac{12}{2\pi} = \dfrac{6}{\pi}\approx 1.9\) cm.
Since the coin completes a full rotation every \(12\) cm, after it coin has rolled \(174\) cm along the line, it would have completed \(\frac{174}{12} = 14.5\) revolutions.
This means the coin has completed \(14\) full revolutions and one-half of a revolution. Thus, the image of \(\pi\) will be upside down.