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r = \frac{b \sqrt{6}}{12}
The volume \( V_{\text{sphere}} \) of the inscribed sphere is:
V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac{b \sqrt{6}}{12} \right)^3 = \frac{4}{3} \pi \frac{6\sqrt{6} b^3}{1728} = \frac{\pi \sqrt{6} b^3}{864}
Now, the ratio of the volume of the sphere to the volume of the tetrahedron is:
\text{Ratio} = \frac{V_{\text{sphere}}}{V_{\text{tetrahedron}}} = \frac{\frac{\pi \sqrt{6} b^3}{864}}{\frac{b^3}{6\sqrt{2}}} = \frac{\pi \sqrt{6} b^3}{864} \cdot \frac{6\sqrt{2}}{b^3} = \frac{\pi \sqrt{6} \cdot 6\sqrt{2}}{864}
\sqrt{6} \cdot \sqrt{2} = \sqrt{12} = 2\sqrt{3}
\text{Ratio} = \frac{\pi \cdot 6 \cdot 2\sqrt{3}}{864} = \frac{12\pi \sqrt{3}}{864} = \frac{\pi \sqrt{3}}{72}
Thus, the ratio is \(\boxed{\frac{\pi \sqrt{3}}{72}}\).
A rectangle with dimensions \( 5 \) cm by \( 12 \) cm is inscribed in a circle. What is the number of centimeters in the circumference of the circle? Express your answer in terms of \( \pi \).
When a rectangle is inscribed in a circle, the diagonal of the rectangle is the diameter of the circle.