$\# 6$ Find the radius $r$ (in inches) for a cone where the height is twice the radius, and the volume is 30.8 cubic inches.

Step 1 ：We are given a cone where the height is twice the radius, and the volume is 30.8 cubic inches. We are asked to find the radius of the cone.

Step 2 ：The volume V of a cone is given by the formula \(V = \frac{1}{3}\pi r^2 h\), where r is the radius and h is the height.

Step 3 ：In this case, we know that \(h = 2r\) and \(V = 30.8\) cubic inches.

Step 4 ：We can substitute \(h = 2r\) into the volume formula to get \(V = \frac{2}{3}\pi r^3\).

Step 5 ：Setting this equal to 30.8 and solving for r, we get three roots: \(r = 2.44998843485815\), \(r = -1.22499421742907 - 2.12175222356523i\), and \(r = -1.22499421742907 + 2.12175222356523i\).

Step 6 ：Since the radius of a cone cannot be a complex number, we discard the complex roots. The remaining root is approximately 2.45, which is the radius of the cone.

Step 7 ：Final Answer: The radius of the cone is approximately \(\boxed{2.45}\) inches.