A solid sphere of radius Xcm is placed in a cylinder of radius 2Xcm and height 2Xcm. The cylinder is then filled with water to the brim and the solid gently withdrawn. Find d volume of d water in d cylinder in cm3
I assume you know how to find the volumes of such solids. Just subtract the volume of the sphere from that of the cylinder. The leftover space holds the water:
π(2x)^2*(2x) - 4/3 πx^3
To find the volume of water in the cylinder after removing the solid sphere, we need to calculate the difference between the volume of the cylinder when it is filled to the brim and the volume of the solid sphere.
1. Volume of the cylinder:
The formula to calculate the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. In this case, the radius of the cylinder is 2X cm and the height is 2X cm. So, the volume of the cylinder is:
V_cylinder = π(2X)^2 * 2X = 4πX^3 cm^3
2. Volume of the solid sphere:
The formula to calculate the volume of a sphere is V = (4/3)πr^3, where r is the radius. In this case, the radius of the solid sphere is X cm. So, the volume of the solid sphere is:
V_sphere = (4/3)πX^3 cm^3
3. Volume of the water in the cylinder:
To find the volume of the water in the cylinder, we subtract the volume of the solid sphere from the volume of the cylinder:
V_water = V_cylinder - V_sphere
= 4πX^3 - (4/3)πX^3
= (12/3)πX^3 - (4/3)πX^3
= (8/3)πX^3 cm^3
Therefore, the volume of the water in the cylinder is (8/3)πX^3 cm^3.