 Hi friends, I am Purva and today we will discuss the following question. A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter of the sphere. Let us begin with the solution now. Now we are given that diameter of the cone is equal to 14 cm. Therefore radius of the cone is equal to 14 upon 2 which is equal to 7 cm and we are also given that height of the cone is equal to 8 cm. Now we know that volume of the cone is equal to 1 upon 3 into pi r square h where we have r is the radius of the cone and h is the height of the cone. Now we mark this as 1, we mark this as 2 and we mark this as 3. So putting 1 and 2 in 3 we get volume of the cone is equal to 1 upon 3 into pi into r square that is radius square. Now here radius of the cone is equal to 7 cm so we have 7 into 7 into h that is height of the cone and this is equal to 8 cm so we have 8 cm. Now this is equal to 392 pi upon 3 cm. We mark this as 4. Now we are given that external diameter of the sphere is equal to 10 cm. Therefore external radius of the sphere is equal to 10 upon 2 which is equal to 5 cm. Now let the internal radius of the sphere be equal to 8 cm and we have to find the internal diameter of the sphere. Now volume of the hollow sphere is equal to 4 upon 3 into pi into r2 cube minus r1 cube where we have r2 is the external radius of the sphere r1 is the internal radius of the sphere. Now putting the value of r2 that is external radius of the sphere and r1 that is the internal radius of the sphere here we get volume of the hollow sphere is equal to 4 upon 3 into pi into r2 cube that is external radius cube. Now here external radius of the sphere is 5 cm so we get 5 cube minus r1 cube that is internal radius cube Now here internal radius of the sphere is x cm so we get minus x cube cubic cm we mark this as 5 Now in the question we are given that a solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. This means that volume of the cone is equal to volume of the hollow sphere this is given to us. Now from 4 we have volume of the cone is equal to 392 pi upon 3 cm and from 5 we have volume of the hollow sphere is equal to 4 upon 3 into pi into 5 cube minus x cube cubic cm So from 4 and 5 we get 392 pi upon 3 is equal to 4 upon 3 into pi into 5 cube minus x cube or we can write this as 392 is equal to 4 into 5 cube minus x cube or 98 is equal to 5 cube minus x cube or we have x cube is equal to 125 minus 98 which is equal to 27 Now this implies x cube is equal to 3 cube or we get x is equal to 3 So we have internal radius of the sphere is equal to 3 cm and hence the internal diameter of the sphere is equal to 3 into 2 which is equal to 6 cm So we have got our answer as 6 cm Hope you have understood the solution Bye and take care