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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 V_{\text{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

📌 For the hemisphere with radius \(3x\):

📌 V_2 = \frac{2}{3}\pi (3x)^3 = \frac{2}{3}\pi (27x^3) = 18\pi x^3

📌 \frac{V_2}{V_1} = \frac{18\pi x^3}{\frac{32}{3}\pi x^3} = \frac{18}{\frac{32}{3}} = 18 \cdot \frac{3}{32} = \frac{54}{32} = \frac{27}{16}

📌 Thus, the ratio is:

📌 \boxed{\frac{27}{16}}

📌 Question:** In a right triangle, the hypotenuse is \(c\) and the radius of the inscribed circle is \(r\). If the area of the triangle is \(A\), express the ratio \(\frac{A}{r^2}\) in terms of \(c\).

📌 For a right triangle with legs \(a\), \(b\), hypotenuse \(c\), and inradius \(r\), the area \(A\) is:

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