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Home » ho = 2a \sin\phi\) represents a sphere of radius \(a\) centered at \((0, 0, a)\). - go-checkin.com

ho = 2a \sin\phi\) represents a sphere of radius \(a\) centered at \((0, 0, a)\). - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 The radius of the circumscribed sphere (circumradius) of a regular tetrahedron is:

📌 R = \frac{s \sqrt{6}}{4}

📌 The volume of the sphere is:

📌 V_{\text{sphere}} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi \left(\frac{s \sqrt{6}}{4}\right)^3 = \frac{4}{3} \pi \frac{6\sqrt{6}}{64} s^3 = \frac{\pi \sqrt{6}}{8} s^3

📌 The ratio of the volume of the tetrahedron to the volume of the sphere is:

📌 \frac{V_{\text{tetrahedron}}}{V_{\text{sphere}}} = \frac{\frac{\sqrt{2}}{12} s^3}{\frac{\pi \sqrt{6}}{8} s^3} = \frac{\sqrt{2}}{12} \cdot \frac{8}{\pi \sqrt{6}} = \frac{2\sqrt{3}}{9\pi}

📌 Thus, the final answer is \(\boxed{\frac{2\sqrt{3}}{9\pi}}\).Question:

📌 A science journalist is writing an article and finds that two identical microscopes cost a total of \$120. What is the cost in dollars of five such microscopes?

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