 Hello and welcome to the session. In this session we will discuss a question which says that a hemispherical repression is cut out from one face of a cubical wooden block of edge 21 cm such that the diameter of the hemisphere is equal to the edge of the cube. Determine the volume and total surface area of the remaining block. Now before starting the solution of this question, we should know some results. First is, volume of the cube is equal to s cube where s is the site of the cube or edge of the cube. Secondly, total surface area of the cube is equal to 6 into side square. Thirdly, volume of a hemisphere is equal to 2 by 3 by r2 where r is the radius of the hemisphere. And joint surface area of the hemisphere is equal to 2 by r square where r is the radius of the hemisphere. And area of the circle is equal to pi r square where r is the radius of the circle and pi is the constant. Now these results will work out as a key idea for solving out this question. And now we will start with the solution. Now in the question, a cubical wooden block with edge 21 cm is given to us and also it is given that a hemispherical depression is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to edge of the cube. That is, diameter of the hemisphere is also 21 cm. Now given edge of the cubical wooden block that is s is equal to 21 cm and the diameter of the hemispherical depression that is the hemisphere is equal to edge of the cubical wooden block which is equal to 21 cm. Now using this result which is given over a key idea, volume of cubical wooden block is equal to size cube which is equal to 21 cm cube which is equal to 9,261 cm2. Now the diameter is equal to 21 cm. Therefore radius is equal to diameter by 2 that is 21 by 2 cm. Now using this result which is given over a key idea, volume of the hemisphere is equal to 2 by 3 by r cube which is equal to 2 by 3 into pi here is 22 by 7 and r is 21 by 2 cm. So it will be 21 by 2 into 21 by 2 into 21 by 2 cm cube which is further equal to now 3 into 7 is 21, 7 is cancelled with 7, 2 into 11 is 22 and here 2 is cancelled with 2. So this will be 4851 by 2 cm cube which is further equal to 2425.5 cm cube. Now first of all we have to find the volume of the remaining block. So here volume of the remaining block will be equal to volume of the cube that is volume of the cubical wooden block minus volume of hemisphere. Now this is the volume of the cube and this is the volume of the hemisphere. So this will be equal to 9261 cm cube minus 2425.5 cm cube which is equal to 6835.5 cm cube. Now we have to find the total surface area of the remaining block. Now this is the edge of the cube and this is the radius of the hemisphere. Now using these results which are given in the key area the total surface area of the cube is equal to 6 into sine square so it will be equal to 6 into 21 cm square which is equal to 6 into 441 cm square which is equal to 2646 cm square. Now the correct surface area of the hemisphere is equal to 2 pi r square which is equal to 2 into 22 by 7 into r square and r here is 21 by 2 cm so it will be 21 by 2 into 21 by 2 cm square. Now here 7 into 3 is 21 2 into 11 is 22 and 2 here will be cancelled with 2 so this is equal to 693 cm square. Now total surface area of the remaining block is equal to total surface area of the cube plus curved surface area of the hemisphere minus area of the circle on the top. That is the total surface area of the remaining block will be the total surface area of the cube plus curved surface area of hemisphere that is this area of the circle on the top that is the circle and the radius of this circle which is on the top is 21 by 2 cm therefore its area will be equal to pi r square and on solving this we will get the area of the circle on the top is 346.5 cm square. Now this is the total surface area of the cube this is the curved surface area of the hemisphere and this is the area of the circle on the top. All these values here this will be equal to 2646 cm square plus 693 cm square minus 346.5 cm square which is further equal to 3339 cm square minus 346.5 cm square which is further equal to 2992.5 cm square So this is the volume of the remaining block and this is the total surface area of the remaining block So this is the solution of the given question and that's all for this session Hope you all have enjoyed the session