# 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?

Result

R =  19.079 cm

#### Solution:

$A = (30^\circ \rightarrow rad) = (30 \cdot \ \dfrac{ \pi }{ 180 } \ ) = 0.523598775598 \ \\ V_{ 1 } = 10 \ l = 10 \cdot \ 1000 \ cm^3 = 10000 \ cm^3 \ \\ \ \\ \cos A = r:R \ \\ \sin A = v:R \ \\ V_{ 2 } = \dfrac{ \pi v }{ 6 } \cdot \ (3r^2 +v^2) \ \\ \ \\ V = V_{ 1 }+V_{ 2 } = \dfrac{ 1 }{ 2 } \cdot \ \dfrac{ 4 }{ 3 } \pi R^3 = \dfrac{ 2 }{ 3 } \pi R^3 \ \\ \ \\ V_{ 2 } = \dfrac{ \pi R \sin A }{ 6 } \cdot \ (3(R \cos A)^2 +(R \sin A)^2) \ \\ \ \\ V_{ 2 } = \dfrac{ \pi R^3 \ \sin A }{ 6 } \cdot \ (3(\cos A)^2 +(\sin A)^2) \ \\ \ \\ \dfrac{ 2 }{ 3 } \pi R^3 = V_{ 1 } + \dfrac{ \pi R^3 \ \sin A }{ 6 } \cdot \ (3(\cos A)^2 +(\sin A)^2) \ \\ \ \\ k = \dfrac{ \pi \cdot \ \sin(A) }{ 6 } \cdot \ (3 \cdot \ (\cos(A))^2 +(\sin(A))^2) = \dfrac{ 3.1416 \cdot \ \sin(0.5236) }{ 6 } \cdot \ (3 \cdot \ (\cos(0.5236))^2 +(\sin(0.5236))^2) \doteq 0.6545 \ \\ \ \\ \dfrac{ 2 }{ 3 } \pi R^3 = V_{ 1 } + k \cdot \ R^3 \ \\ \ \\ R = \sqrt{ \dfrac{ V_{ 1 } }{ \dfrac{ 2 }{ 3 } \cdot \ \pi - k } } = \sqrt{ \dfrac{ 10000 }{ \dfrac{ 2 }{ 3 } \cdot \ 3.1416 - 0.6545 } } \doteq 19.079 \ cm \ \\ \ \\ \ \\ V = \dfrac{ 2 }{ 3 } \cdot \ \pi \cdot \ R^3 = \dfrac{ 2 }{ 3 } \cdot \ 3.1416 \cdot \ 19.079^3 \doteq 14545.4545 \ cm^3 \ \\ r = R \cdot \ \cos(A) = 19.079 \cdot \ \cos(0.5236) \doteq 16.5229 \ cm \ \\ v = R \cdot \ \sin(A) = 19.079 \cdot \ \sin(0.5236) \doteq 9.5395 \ cm \ \\ V_{ 2 } = \dfrac{ \pi \cdot \ v }{ 6 } \cdot \ (3 \cdot \ r^2 +v^2) = \dfrac{ 3.1416 \cdot \ 9.5395 }{ 6 } \cdot \ (3 \cdot \ 16.5229^2 +9.5395^2) = \dfrac{ 50000 }{ 11 } \doteq 4545.4545 \ cm^3 \ \\ \ \\ V_{ 8 } = V-V_{ 2 } = 14545.4545-4545.4545 = 10000 \ cm^3 \ \\ V_{ 8 } = V_{ 1 } \ \\ \ \\ \ \\ R = 19.079 \doteq 19.079 = 19.079 \ \text { cm }$ Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

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