20 identical solid hemispheres of radius 3 cm
20 identical solid hemispheres of radius 3 cm are melted to form a big solid hemisphere. What is the radius (in cm) of the biggest hemisphere formed, if 20% of the solid is wasted during the process?
3 cm त्रिज्या वाले 20 एकसमान ठोस अर्धगोलों को पिघलाकर एक बड़ा ठोस अर्धगोला बनाया जाता है। यदि इस प्रक्रिया के दौरान 20% ठोस बर्बाद जाता है, तो बनाए गए सबसे बड़े अर्धगोल की त्रिज्या (cm में) कितनी हैं?
Detailed Solution & Logic
6\(\sqrt[3]{2}\)
Volume of small hemisphere = $ \frac{2}{3}\pi r^3 = \frac{2}{3}\pi (3)^3 = 18\pi $
Volume of 20 hemispheres = $ 20 \cdot 18\pi = 360\pi $
After 20% wastage: $ V_\text{big} = 0.8 \cdot 360\pi = 288\pi $
Volume of big hemisphere =$ \frac{2}{3}\pi R^3 = 288\pi $
$ R^3 = \frac{288 \pi \cdot 3}{2 \pi} = 432 = 216 \cdot 2 = 6^3 \cdot 2 $
$ R = \sqrt[3]{6^3 \cdot 2} = 6 \sqrt[3]{2} $
${6 \sqrt[3]{2}} $
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