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📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 V_s = \frac{4}{3} \pi x^3

📌 The volume \( V_h \) of a hemisphere with radius \( 3x \) is half the volume of a sphere with the same radius:

📌 V_h = \frac{1}{2} \times \frac{4}{3} \pi (3x)^3 = \frac{1}{2} \times \frac{4}{3} \pi \times 27x^3 = \frac{54}{3} \pi x^3 = 18 \pi x^3

📌 \text{Ratio} = \frac{V_s}{V_h} = \frac{\frac{4}{3} \pi x^3}{18 \pi x^3} = \frac{4}{3} \times \frac{1}{18} = \frac{4}{54} = \frac{2}{27}

📌 \boxed{\frac{2}{27}}

📌 Question:** A retired scientist is mentoring students on geometric proportions. An equilateral triangle has an area of \( 36\sqrt{3} \, \text{cm}^2 \). If each side is reduced by 2 cm, by how many square centimeters does the area decrease?

📌 Set the area equal to \( 36\sqrt{3} \):

📌 \frac{1}{4} s^2 = 36 \Rightarrow s^2 = 144 \Rightarrow s = 12 \, \text{cm}

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