The hemisphere

The hemisphere container is filled with water. What is the radius of the container when 10 liters of water pour from it when tilted 30 degrees?

Correct answer:

R = 19.079 cm

Step-by-step explanation:

A = 30 ° \to rad = 30 ° \cdot \frac{π}{1 8 0} = 30 ° \cdot \frac{3 . 1 4 1 5 9 2 6}{1 8 0} = 0.5236 = π / 6 V_{1} = 10 l \to cm^{3} = 10 \cdot 1000 cm^{3} = 10000 cm^{3} cos A = r : R sin A = v : R V_{2} = \frac{π v}{6} \cdot (3 r^{2} + v^{2}) V = V_{1} + V_{2} = \frac{1}{2} \cdot \frac{4}{3} π R^{3} = \frac{2}{3} π R^{3} V_{2} = \frac{π R sin A}{6} \cdot (3 (R cos A)^{2} + (R sin A)^{2}) V_{2} = \frac{π R ^{3} sin A}{6} \cdot (3 (cos A)^{2} + (sin A)^{2}) \frac{2}{3} π R^{3} = V_{1} + \frac{π R ^{3} sin A}{6} \cdot (3 (cos A)^{2} + (sin A)^{2}) k = \frac{π \cdot sin ( A )}{6} \cdot (3 \cdot (cos (A))^{2} + (sin (A))^{2}) = \frac{3 . 1 4 1 6 \cdot sin 0 . 5 2 3 6}{6} \cdot (3 \cdot (cos 0.5236)^{2} + (sin 0.5236)^{2}) ≐ 0.6545 \frac{2}{3} π R^{3} = V_{1} + k \cdot R^{3} R = 3 \frac{V _{1}}{\frac{2}{3} \cdot π - k} = 3 \frac{1 0 0 0 0}{\frac{2}{3} \cdot 3 . 1 4 1 6 - 0 . 6 5 4 5} ≐ 19.079 cm V = \frac{2}{3} \cdot π \cdot R^{3} = \frac{2}{3} \cdot 3.1416 \cdot 19.07 9^{3} ≐ 14545.4545 cm^{3} r = R \cdot cos (A) = 19.079 \cdot cos 0.5236 ≐ 16.5229 cm v = R \cdot sin (A) = 19.079 \cdot sin 0.5236 ≐ 9.5395 cm V_{2} = \frac{π \cdot v}{6} \cdot (3 \cdot r^{2} + v^{2}) = \frac{3 . 1 4 1 6 \cdot 9 . 5 3 9 5}{6} \cdot (3 \cdot 16.522 9^{2} + 9.539 5^{2}) = \frac{5 0 0 0 0}{1 1} ≐ 4545.4545 cm^{3} V_{8} = V - V_{2} = \frac{1 6 0 0 0 0}{1 1} - \frac{5 0 0 0 0}{1 1} = \frac{1 6 0 0 0 0 - 5 0 0 0 0}{1 1} = \frac{1 1 0 0 0 0}{1 1} = 10000 cm^{3} V_{8} = V_{1} R = 19.079 = 19.079 cm

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