 Hello and welcome to the session. In this session we discuss the following question which says a solid is in form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 3.5 centimeters and the height of the cone is 4 centimeters. The solid is placed in a cylindrical tub full of water in such a way that the whole solid is submerged in water. If the radius of the cylinder is 5 centimeters and its height is 10.5 centimeters, find the volume of the water left in the cylindrical tub. So we have a solid which is in the form of a right circular cone that is mounted on a hemisphere and it is given that the solid is placed in the cylindrical tub which is full of water such that the whole solid is submerged in water. Now we have to find out the volume of water which is left in the given cylindrical tub. First of all let's recall some formulae. We have volume of a cylinder is equal to pi r square h. This r is the radius of the base of the cylinder. h is the height of the cylinder. Then volume of a cone is given by 1 by 3 pi r square h where again this r is the radius of the base of the cone. h is the height of the cone. Then volume of a hemisphere is equal to 2 by 3 pi r cube. This r is the radius of the base of the hemisphere. So this is the key idea that we use in this question. Now let's see the solution. Now this is the solid given to us where this ACB is the conical part of the given solid and AO dash B is the hemispherical part of the solid. We are given that the height of the cone given by h is equal to 4 centimeters. Then radius of the base of the cone given by r is equal to 3.5 centimeters. Since we are given the radius of the hemisphere and that would be the same as the radius of the base of the cone. So as the radius of the hemisphere is given as 3.5 centimeters. So radius of the base of the cone ACB is also 3.5 centimeters. So now volume of the cone ACB given by V would be equal to 1 by 3 pi r square h. Now we can substitute the values for r and h to get the volume of the cone ACB. This is equal to 1 by 3 pi into 3.5 square into 4 centimeter cube is the volume of the cone. Then we have the radius of the base of the hemisphere AO dash B given by r dash is equal to 3.5 centimeters. Then the volume of the hemisphere AO dash B given by V dash is equal to 2 by 3 pi r dash cube. Now we have the value for r dash. So we substitute this value of r dash and so this volume V dash would be equal to 2 by 3 pi into 3.5 cube centimeter cube is the volume V dash of the hemisphere AO dash B. Now as you know the solid is combination of the cone and the hemisphere. So volume of the solid would be equal to the volume of the cone which is V plus the volume of the hemisphere which is V dash and so this is equal to 1 by 3 into pi into 3.5 whole square into 4 plus 2 by 3 pi into 3.5 whole cube centimeter cube is the volume of the solid. This is further equal to 1 by 3 into pi into 3.5 whole square this whole into 4 plus 2 into 3.5 centimeter cube and so this is equal to 1 by 3 into pi into 12.25 into 11 centimeter cube. This further is equal to 134.75 pi upon 3 centimeter cube. So we have now got the volume of the solid. Next we are also given the dimensions of the cylindrical tub which is full of water and in which the solid is submerged. So we have the radius of the cylinder capital R is equal to 5 centimeters then the height of the cylinder capital H is equal to 10.5 centimeters. So the volume of the cylinder given by V double dash is equal to pi R square into H this is equal to pi into 5 square into 10.5 centimeter cube this is equal to 262.5 pi centimeter cube is the volume of the cylinder that is the volume of the cylindrical tub. Now it's given in the question that the whole solid is submerged in the water which is in the cylindrical tub. So we say when the solid is submerged in the cylindrical tub then the volume of water that flows out of the cylinder is equal to the volume of the solid and we have to find out the volume of water left in the cylindrical tub. So volume of the water left in the cylinder is equal to the volume of the cylinder minus the volume of the water that flows out of the cylinder. Now since volume of the water that flows out of the cylinder is equal to the volume of the solid so this would be equal to that is the volume of water left in the cylinder is equal to volume of the cylinder minus the volume of the solid. Now we have the volume of the cylinder that is V double dash as 262.5 pi centimeter cube and the volume of the solid is 134.75 upon 3 pi centimeter cubed. So this is equal to 262.5 pi minus 134.75 upon 3 pi centimeter cube is the volume of the water left in the cylinder. This further is equal to 787.5 minus 134.75 this whole upon 3 into pi centimeters cube. So further we have 652.75 upon 3 into pi centimeter cube and this is equal to 217.583 pi centimeter cube. Now we put the value of pi as 3.14 and this is equal to 683.21 centimeter cube that is we have the volume of the water left in the cylinder is equal to 683.21 centimeter cube. So this is our final answer. This completes the session. Hope you have understood the solution of this question.