a hemispherical bowl of radius r cm is filled with water.a sphere of radius r/2 cm is immersed in it.show that the volume of water displaces is pie r square/2 c.c.
volume of sphere = (4/3)πr^3
water displaced by a sphere of radius r/2
= (4/3)π(r^3/8
= πr^3 /6
To prove this, we can calculate the volume of the hemispherical bowl before and after the sphere is immersed in it.
Let's start with the volume of the hemispherical bowl:
The volume of a hemisphere is given by the formula V = (2/3)πr^3, where r is the radius of the hemisphere.
So, the volume of the hemispherical bowl is V1 = (2/3)πr^3.
Now, let's consider the volume of water displaced after the sphere is immersed in it:
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.
In this case, the radius of the sphere is r/2, so the volume of the sphere is V2 = (4/3)π(r/2)^3 = (4/3)π(r^3/8) = (1/6)πr^3.
Since the sphere is completely immersed in the hemispherical bowl, the volume of water displaced is the volume of the sphere, which is V2 = (1/6)πr^3.
Therefore, the volume of water displaced is given by V1 - V2 = (2/3)πr^3 - (1/6)πr^3 = (4/6)πr^3 - (1/6)πr^3 = (3/6)πr^3 = (1/2)πr^3.
Hence, the volume of water displaced is (1/2)πr^3, which is the required solution.
Please note that we used the relationship between the radius of the sphere and the radius of the hemisphere to calculate the volume of water displaced.