 Hello and welcome to the session. In this session we discuss the following question which says from a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm is hollowed out. Find the volume of the remaining solid correct to two places of decimal also find the total surface area of the remaining solid. Before moving on to the solution let's recall the formula to find the volume of cylinder this is equal to pi r square h then the formula to find volume of cone is equal to 1 by 3 pi r square h next we have the formula to find the curved surface area of cylinder this is equal to 2 pi r h then curved surface area of cone is given by pi r l this is the key idea that we use in this question. Let's proceed with the solution now. Consider this figure in which we have a cylinder and a conical cavity is hollowed out of the cylinder we have the height of the cylinder is equal to 8 cm then the radius of the cylinder is equal to 6 cm so let's find out the volume of the cylinder this is equal to pi r square h that is equal to pi into r that is the radius of the cylinder which is 6 cm so 6 square into height that is 8 and so this is equal to 288 pi centimeter cube is the volume of the cylinder now we are given that a conical cavity is hollowed out of the cylinder and we are given the height of the cone be equal to 8 cm and the base radius of the cone is equal to 6 cm so now volume of the cone is equal to 1 by 3 pi r square h where r is the base radius of the cone and h is the height of the cone we substitute the respective values and so this is equal to 1 by 3 into pi into 6 square into 8 and this is further equal to 96 pi centimeter cube is the volume of the cone we have to find the volume of the remaining solid so volume of the remaining solid is equal to volume of the cylinder minus volume of the cone this is equal to volume of cylinder that is 288 pi centimeter cube minus volume of cone which is 96 pi centimeter cube so this is equal to pi into 288 minus 96 centimeter cube which is equal to 192 pi centimeter cube we put the value for pi as 3.1416 and this comes out to be equal to 603.19 centimeter cube approximately so this is the volume of the remaining solid correct up to 2 places of decimal next we need to find the total surface area of the remaining solid and this is equal to the curved surface area of the cylinder the curved surface area of the cone plus the area of the circle that is pi r square now the curved surface area of the cylinder is given by the formula 2 pi r h plus the curved surface area of the cone is given by the formula pi r l plus pi r square now this is equal to pi into 2 r h plus r l plus r square now this l is the planned height of the cone which is given by square root of r square plus h square that is equal to square root of r that is the base radius of the cone which is 6 square plus the height of the cone that is 8 square and this is equal to square root of 36 plus 64 which is equal to square root of 100 and this is equal to 10 thus we get the planned height of the cone is equal to 10 centimeters so now this is further equal to pi into 2 into r that is the radius of the cylinder which is 6 into height of the cylinder h that is 8 plus radius of the cone which is 6 into planned height l of the cone that is 10 plus r square that is the radius of the circle which is 6 so 6 square and this is further equal to pi into 96 plus 60 plus 36 further we get pi into 192 now we substitute the value for pi as 3.1416 into 192 which is equal to 603.19 centimeters square approximately so we get the total surface area of the remaining solid is equal to 603.19 centimeters square approximately so this completes the session hope you have understood the solution of this question