# 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:

$A = (30^\circ \rightarrow rad) = (30 \cdot \ \dfrac{ \pi }{ 180 } \ ) = 0.523598775598 \ \\ V_{ 1 } = 10 \ l = 10 \cdot \ 1000 \ cm^3 = 10000 \ cm^3 \ \\ \ \\ \cos A = r:R \ \\ \sin A = v:R \ \\ V_{ 2 } = \dfrac{ \pi v }{ 6 } \cdot \ (3r^2 +v^2) \ \\ \ \\ V = V_{ 1 }+V_{ 2 } = \dfrac{ 1 }{ 2 } \cdot \ \dfrac{ 4 }{ 3 } \pi R^3 = \dfrac{ 2 }{ 3 } \pi R^3 \ \\ \ \\ V_{ 2 } = \dfrac{ \pi R \sin A }{ 6 } \cdot \ (3(R \cos A)^2 +(R \sin A)^2) \ \\ \ \\ V_{ 2 } = \dfrac{ \pi R^3 \ \sin A }{ 6 } \cdot \ (3(\cos A)^2 +(\sin A)^2) \ \\ \ \\ \dfrac{ 2 }{ 3 } \pi R^3 = V_{ 1 } + \dfrac{ \pi R^3 \ \sin A }{ 6 } \cdot \ (3(\cos A)^2 +(\sin A)^2) \ \\ \ \\ k = \dfrac{ \pi \cdot \ \sin(A) }{ 6 } \cdot \ (3 \cdot \ (\cos(A))^2 +(\sin(A))^2) = \dfrac{ 3.1416 \cdot \ \sin(0.5236) }{ 6 } \cdot \ (3 \cdot \ (\cos(0.5236))^2 +(\sin(0.5236))^2) \doteq 0.6545 \ \\ \ \\ \dfrac{ 2 }{ 3 } \pi R^3 = V_{ 1 } + k \cdot \ R^3 \ \\ \ \\ R = \sqrt[3]{ \dfrac{ V_{ 1 } }{ \dfrac{ 2 }{ 3 } \cdot \ \pi - k } } = \sqrt[3]{ \dfrac{ 10000 }{ \dfrac{ 2 }{ 3 } \cdot \ 3.1416 - 0.6545 } } \doteq 19.079 \ cm \ \\ \ \\ \ \\ V = \dfrac{ 2 }{ 3 } \cdot \ \pi \cdot \ R^3 = \dfrac{ 2 }{ 3 } \cdot \ 3.1416 \cdot \ 19.079^3 \doteq 14545.4545 \ cm^3 \ \\ r = R \cdot \ \cos(A) = 19.079 \cdot \ \cos(0.5236) \doteq 16.5229 \ cm \ \\ v = R \cdot \ \sin(A) = 19.079 \cdot \ \sin(0.5236) \doteq 9.5395 \ cm \ \\ V_{ 2 } = \dfrac{ \pi \cdot \ v }{ 6 } \cdot \ (3 \cdot \ r^2 +v^2) = \dfrac{ 3.1416 \cdot \ 9.5395 }{ 6 } \cdot \ (3 \cdot \ 16.5229^2 +9.5395^2) = \dfrac{ 50000 }{ 11 } \doteq 4545.4545 \ cm^3 \ \\ \ \\ V_{ 8 } = V-V_{ 2 } = 14545.4545-4545.4545 = 10000 \ cm^3 \ \\ V_{ 8 } = V_{ 1 } \ \\ \ \\ \ \\ R = 19.079 \doteq 19.079 = 19.079 \ \text { cm }$

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!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

#### Following knowledge from mathematics are needed to solve this word math problem:

Do you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation? Do you know the volume and unit volume, and want to convert volume units? Pythagorean theorem is the base for the right triangle calculator. See also our trigonometric triangle calculator. Try conversion angle units angle degrees, minutes, seconds, radians, grads.

## Next similar math problems:

1. Pool
If water flows into the pool by two inlets, fill the whole for 8 hours. The first inlet filled pool 6 hour longer than second. How long pool take to fill with two inlets separately?
2. TV transmitter
The volume of water in the rectangular swimming pool is 6998.4 hectoliters. The promotional leaflet states that if we wanted all the pool water to flow into a regular quadrangle with a base edge equal to the average depth of the pool, the prism would have.
3. Trapezoid MO
The rectangular trapezoid ABCD with right angle at point B, |AC| = 12, |CD| = 8, diagonals are perpendicular to each other. Calculate the perimeter and area of ​​the trapezoid.
4. Cube in a sphere
The cube is inscribed in a sphere with volume 7253 cm3. Determine the length of the edges of a cube.
5. Axial section
Axial section of the cone is an equilateral triangle with area 168 cm2. Calculate the volume of the cone.
6. Cuboid
Cuboid with edge a=6 cm and body diagonal u=31 cm has volume V=900 cm3. Calculate the length of the other edges.
7. Logic
A man can drink a barrel of water for 26 days, woman for 48 days. How many days will a barrel last between them?
8. River
Calculate how many promiles river Dunaj average falls, if on section long 957 km flowing water from 1454 m AMSL to 101 m AMSL.
9. Circle chord
What is the length d of the chord circle of diameter 51 mm, if the distance from the center circle is 19 mm?
10. Tetrahedral pyramid
What is the surface of a regular tetrahedral (four-sided) pyramid if the base edge a=16 and height v=16?
11. Rectangle
The rectangle is 21 cm long and 38 cm wide. Determine the radius of the circle circumscribing rectangle.
12. Bonus
Gross wage was 527 EUR including 16% bonus. How many EUR were bonuses?
13. Beer
After three 10° beers consumed in a short time, there is 5.6 g of alcohol in 6 kg adult human blood. How much is it per mille?
14. Trigonometric functions
In right triangle is: ? Determine the value of s and c: ? ?
15. Clock
How many times a day hands on a clock overlap?
16. Server
Calculate how many average minutes a year is a webserver is unavailable, the availability is 99.99%.
17. Triangle SAS
Calculate the area and perimeter of the triangle, if the two sides are 51 cm and 110 cm long and angle them clamped is 130 °.