 Hello and welcome to the session. Let us discuss the following question. Question says, a container opened from the top and made of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of the milk which can completely fill the container at the rate of Rs 20 per litre. Also, find the cost of metal sheet used to make the container if it costs Rs 8 per 100 cm square. Take pi is equal to 3.14. First of all, let us understand that curved surface area of a frustum of a cone is equal to pi multiplied by L multiplied by r1 plus r2 where L is land height of the frustum and r1 and r2 are radii of 2 circular ends. Also, volume of frustum of a cone is equal to 1 upon 3 pi H multiplied by r1 square plus r2 square plus r1 r2. Now here, H is the height of the frustum and r1 and r2 are radii of 2 circular ends of the frustum. Now let us know formula for land height of the frustum. Land height of the frustum that is L is equal to square root of H square plus r1 minus r2 whole square. Now in all these formulas, L represents land height of frustum, H represents height of the frustum, r1 and r2 are radii of 2 circular ends of the frustum. Now we will use all these formulas as our key idea to solve the given question. Let us now start with the solution. We are given that height of the frustum of a cone is equal to 16 centimeter, so we can write height of frustum that is H is equal to 16 centimeters and we are also given that radius of upper circular end that is r1 is equal to 20 centimeters, radius of lower circular end that is r2 is equal to 8 centimeters. Clearly we can see this is a container where radius of the upper circular end is equal to 20 centimeters, radius of the lower circular end is equal to 8 centimeters and height of the container is equal to 16 centimeters. Now we have to find the cost of the milk which can completely fill the container at the rate of rupees 20 per liter. So first of all we will find capacity of the container or we can say we will find volume of the container. We know amount of milk that completely fills the container is equal to volume of the container or we can say it is equal to volume of the frustum. We know container is in the shape of frustum of a cone. From key idea we know volume of the frustum is equal to 1 upon 3 pi H multiplied by square of r1 plus square of r2 plus r1 r2 where H is the height of the frustum and r1 and r2 are radii of two circular ends. Now this is further equal to 1 upon 3 multiplied by 3.14 multiplied by 16 multiplied by 20 square plus 8 square plus 20 multiplied by 8. We are given that value of pi is equal to 3.14, H is equal to 16 centimeters, r1 is equal to 20 centimeters, r2 is equal to 8 centimeters. So substituting their corresponding values in this expression we get this expression. Now simplifying further we get 1 upon 3 multiplied by 3.14 multiplied by 16 multiplied by 400 plus 64 plus 160 centimeter cube square of 20 is 400 square of 8 is 64 and 20 multiplied by 8 is equal to 160. Now adding these three terms we get 624 so we can write 1 upon 3 multiplied by 3.14 multiplied by 16 multiplied by 624 centimeter cube. Now this is further equal to 31349.76 upon 3 centimeter cube. Now this can be further written as 10449.92 centimeter cube. Now we know 1000 centimeter cube is equal to 1 liter so this can be written as 10449.92 upon 1000 liters. 1 centimeter cube is equal to 1 upon 1000 liters so 10449.92 centimeter cube is equal to 10449.92 upon 1000 liters. Now this value can be further written as 10449.92 liters or we can write it as 10.45 liters approximately. Rounding of this value up to two places of decimals we get this value. Now we get volume of milk that completely fills the container is equal to 10.45 liters. Now we have to find cost of milk that completely fills the container at the rate of rupees 20 per liters. So cost of milk that completely fills the container at the rate of rupees 20 per liter is equal to 10.45 multiplied by 20 rupees. Now multiplying these two terms we get rupees 209. So we get cost of milk which can completely fill the container is equal to rupees 209. Now this completes the first part of the question. Let us now start with the second part. We have to find the cost of metal sheet used. Now we will find out surface area of this given container or we can say we will find out surface area of the frustum. Now clearly we can see surface area of this frustum is equal to curved surface area of the frustum plus area of the lower circular end. We know in this container top is open or we can say top of this frustum is open. So we can write surface area of given frustum is equal to curved surface area of frustum plus area of lower circular end. From key idea we know curved surface area of this frustum is equal to pi L multiplied by r1 plus r2 and area of this lower circular end is equal to pi multiplied by square of r2. Now here L represents the slant height and r1 and r2 are radii of upper and lower circular ends. Now first of all we will find out slant height of the given frustum. From key idea we know slant height is equal to r1 minus r2 whole square plus square of h. Now substituting corresponding values of r1, r2 and h in this expression we get slant height is equal to square root of 20 minus 8 whole square plus square of 16. Now this is further equal to square of 12 plus square of 16 20 minus 8 is equal to 12. So we can write here 12 square. Now this implies slant height is equal to 144 plus 256. Square of 12 is 144 and square of 16 is 256. Now adding these two terms we get L is equal to square root of 400. Now this further implies L is equal to 20 centimeters. Now we know total surface area of frustum is equal to pi L multiplied by r1 plus r2 plus pi multiplied by square of r2. Now substituting corresponding values of L r1 and r2 in this expression we get pi multiplied by 20 multiplied by 20 plus 8 plus pi multiplied by square of 8. Now clearly we can see pi is common in both these terms. So we can write this expression as pi multiplied by 20 multiplied by 28 we know 20 plus 8 is equal to 28 plus 64 we know square of 8 is equal to 64. Now we will substitute 3.14 for pi and we get 3.14 multiplied by 560 plus 64 centimeters square. Now adding these two terms we get 624. So we can write this expression as 3.14 multiplied by 624 centimeters square. Now this is further equal to 1959.36 centimeters square. Now total metal sheet used to make the container is equal to total surface area of the frustum that is 1959.36 centimeters square. Now we will find cost of metal sheet used at the rate of rupees 8 per 100 centimeters square it is equal to 1959.36 multiplied by 8 upon 100 rupees we are given cost of 100 centimeters square of metal sheet is equal to rupees 8. So this implies cost of 1 centimeter square of metal sheet is equal to rupees 8 upon 100 and we will multiply this cost by total area of the sheet used to find cost of metal sheet used. Now this expression can be further written as rupees 19.59 multiplied by 8 dividing this term by 100 and rounding it off up to two places of decimals we get 19.59. Now multiplying these two terms we get rupees 156.72. So our required answer is cost of metal sheet used to make the container is equal to rupees 156.72. So our required answer is cost of milk is rupees 209 and the cost of metal sheet is rupees 156.72. This completes the session hope you understood the solution take care and keep smiling.