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.
To prove that the volume of water displaced is πr^2/2 cc when a sphere of radius r/2 cm is immersed in a hemispherical bowl of radius r cm, we can follow these steps:
Step 1: Find the volume of the hemispherical bowl.
The volume of a hemisphere can be calculated using the formula V = (2/3)πr^3.
Thus, the volume of the hemispherical bowl is (2/3)πr^3.
Step 2: Find the volume of the immersed sphere.
The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere.
So, the volume of the immersed sphere is (4/3)π(r/2)^3, which simplifies to (1/6)πr^3.
Step 3: Calculate the volume of water displaced.
The volume of water displaced is equal to the volume of the hemispherical bowl minus the volume of the immersed sphere.
Therefore, the volume of water displaced is (2/3)πr^3 - (1/6)πr^3, which simplifies to (4/6 - 1/6)πr^3 or, simply, (3/6)πr^3.
Step 4: Simplify the expression.
The fraction 3/6 can be simplified to 1/2.
So, the volume of water displaced is (1/2)πr^3, which is equal to πr^2/2 cc.
Hence, we have proved that the volume of water displaced when a sphere of radius r/2 cm is immersed in a hemispherical bowl of radius r cm is πr^2/2 cc.
To show that the volume of water displaced is πr^2/2 cc, we need to calculate the volume of the sphere and subtract it from the volume of the hemisphere.
First, let's find the volumes of the hemisphere and the sphere:
1. Volume of the hemisphere:
The volume of a hemisphere is given by:
V_hemisphere = (2/3)πr^3
2. Volume of the sphere:
The radius of the sphere is r/2, so the volume of the sphere is:
V_sphere = (4/3)π(r/2)^3 = (4/3)π(r^3/8) = πr^3/6
Now, we subtract the volume of the sphere from the volume of the hemisphere to find the volume of water displaced:
Volume_displaced = V_hemisphere - V_sphere
= (2/3)πr^3 - πr^3/6
= (4/6)πr^3 - πr^3/6
= (3/6)πr^3 - (1/6)πr^3
= (2/6)πr^3
= (1/3)πr^3
Since we need the volume in cc, we can rewrite it as:
Volume_displaced = (1/3)πr^3 cc
Now let's compare this with the given expression:
πr^2/2 cc = (1/2)πr^2 cc
To show that they are equal, we need to divide both sides by r and compare:
(1/3)πr^3 cc / r = (1/2)πr^2 cc / r
Simplifying, we get:
(1/3)πr^2 cc = (1/2)πr^2 cc
Since the left-hand side is equal to the right-hand side, we have shown that the volume of water displaced is indeed πr^2/2 cc.