 Hello and welcome to the session I am Deepika here. Let's discuss the question which says a cylindrical bucket 32 cm high and with radius of base 18 cm is filled with sand. This bucket is amputated on the ground and the conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap. Now we know that volume of a cylinder is equal to pi r square h where r is the radius h is the height of the cylinder. Volume of a cone is equal to 1 by 3 pi r square h. Again r is the radius and h is the height of the cone and slant height of a cone is equal to l which is equal to under root of h square plus r square. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now we are given a cylindrical bucket 32 cm high. So this is 32 cm and with radius of base 18 cm is filled with sand. Now this bucket is amputated on the conical heap of sand is formed. Now the height of the conical heap is 24 cm. So this height is 24 cm. The radius and slant height of the heap. So given height of the cylinder is equal to 32 cm. Radius of the cylinder is 18 cm. Therefore volume of sand in cylinder is equal to volume of cylinder. So this is given by the formula pi r square h. So this is equal to pi is equal to 22 upon 7 into r square r is 18 cm that is 18 into 18 into height is 32 cm. So volume of sand in cylinder is 22 upon 7 into 18 into 18 into 32 cm cube. Now this volume of sand and we have to find the radius of the conical heap. Therefore volume of equal to volume of sand in conical heap. Now volume of sand in cylinder is 22 upon 7 into 18 into 18 into 32 cm cube. And this is equal to volume of sand in conical heap. And we know the formula for volume of a cone. This is 1 by 3 into pi take pi is 22 upon 7 into r square into h. Now the height of the conical heap is 24 cm. Now we will solve this equation. This implies r square is equal to 18 into 18 into 32 upon this implies r square is equal to 18 into 18 into. Now 4 can be written as 2 into 2. So this implies is equal to under root of 18 into 18 into 2 into 2. So therefore r is equal to 36 cm and this is the radius of the conical heap. Now we will find the sand height of the conical heap is given by r is equal to that is the radius of this conical heap is 36 cm. We will find the sand height of the conical heap. Now sand height is given by the formula under root of h square plus r square. So this is equal to now h is 24 cm. So this is 24 square plus 36 square because radius is 36 cm. Now this is equal to now 24 square is 576 plus 36 square is 1296. So on adding we get this is equal to under root of 1872. Now let us factorize 1872 we get this is equal to under root of 4 into 4 into 3 into 3 into 13. So under root of 4 into 4 into 3 into 3 into 13 so this is equal to 4 into 3 into root 13 of the conical heap equal to 12 cm. Hence the answer for the above question is the radius of the conical heap is 36 cm. The height of the conical heap is 12 root 13 cm. I hope the solution is clear to you. Bye and take care.