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Home » t = \frac{4 \pm \sqrt4}6 - High Altitude Science

t = \frac{4 \pm \sqrt4}6 - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Thus, \(m^3 + n^3 = \boxed{133}\).**Question:

📌 A cube has a side length of \( a \) units. A sphere is inscribed within the cube such that it touches all six faces of the cube. What is the ratio of the volume of the sphere to the volume of the cube?

📌 The volume of the cube is given by the formula:

📌 V_{\text{cube}} = a^3

📌 Since the sphere is inscribed within the cube, its diameter is equal to the side length of the cube, \( a \). Therefore, the radius \( r \) of the sphere is:

📌 r = \frac{a}{2}

📌 The volume of the sphere is given by the formula:

📌 V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac{a}{2} \right)^3 = \frac{4}{3} \pi \frac{a^3}{8} = \frac{\pi a^3}{6}

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