A spherical vessel of diameter $$14$$ cm and height $$42$$ cm is fixed symmetrically inside a similar vessel of diameter $$16$$ cm and height $$42$$ cm. The total space between the two vessels is filled with cork dust for heat insulation purposes. How many cubic centimeters of cork dust will be required?

Question

Vedantu Content Team · Accepted Answer

Hint: A sphere is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.$${\text{Volume of sphere }} = \dfrac{4}{3}\pi {r^3}$$$${\text{Volume of cylinder}} = \pi {r^2}h$$ Complete answer:  To find the volume of cork dust we have to subtract the volume of the inner cylinder from the volume of the outer cylinder.  $${\text{Volume of cork dust }} = {\text{ Volume of outer cylinder }} - {\text{ volume of inner cylinder}}$$$$V = \pi \left( {{\text{r}}_2^2 - {\text{r}}_1^2} \right){\text{h}}$$$${\text{V}} = \pi \left( {{8^2} - {{\text{7}}^2}} \right){\text{42}}$$$$ = {\text{198}}0{\text{c}}{{\text{m}}^3}$$Hence, the volume of cork dust is $${\text{198}}0{\text{c}}{{\text{m}}^3}$$ Note:  Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point in a three-dimensional space. This distance r is the radius of the ball, which is made up of all points with a distance less than (or, for a closed ball, less than or equal to) r from the given point, which is the center of the mathematical ball. These are also referred to as the radius and center of the sphere, respectively.