 Hi and welcome to the session. Let us discuss the following question. Question says a matric right circular cone 20 centimeters high and whose vertical angle is 60 degrees is cut into two parts at the middle of its height by a plane parallel to its space. If the frustum so obtained be drawn into a wire of diameter 1 upon 16 centimeters, find the length of the wire. First of all let us understand that volume of frustum is equal to 1 upon 3 pi h multiplied by r1 square plus r2 square plus r1 r2. Here r1 and r2 are radii of two circular ends of the given frustum and h is the height of the frustum. Also volume of cylinder is equal to pi r square h where r and h are the radius and height of the cylinder. Now we will use these formulas as our key idea to solve the given question. Let us now start with the solution. Now we are given a matric right circular cone 20 centimeters high and whose vertical angle is 60 degrees. So this angle is 60 degrees. Now we can write let a v c be the matric right circular cone whose height h is equal to 20 centimeters. Now we are given that this cone is cut by a plane parallel to its base at the point o such that a o is equal to o f. This implies o is the midpoint of a f. So we can write this cone is cut by a plane parallel to its base at point o such that a o is equal to o f. Now after cutting this cone we are left with this frustum b c d e. Now let us assume that r1 and r2 are radii of two circular ends of the given frustum. So we can write od is equal to oe is equal to r1 and cf is equal to bf is equal to r2. Now we will write let r1 and r2 are radii of two circular ends of the given frustum b c d e. Therefore r1 is equal to od is equal to oe and r2 is equal to cf is equal to bf. We are also given that vertical angle of the cone that is angle ead is equal to 60 degrees. Now we are given in the question that this frustum b c d e is drawn into a wire of diameter 1 upon 16 centimeters. Now first of all we will find out volume of this frustum. It has been drawn into a wire so volume of wire is equal to volume of frustum. Now to find the volume of frustum we will first find out value of r1 and r2. First of all let us consider these two triangles triangle aoe and triangle aod in triangle aoe and triangle aod. Aoe is equal to aoe it is common in both the triangles. Oe is equal to od both are radii of same circle and angle aoe is equal to angle aod as they are 90 degrees each. We know aoe represents the height of this cone and height is always perpendicular to the base. So angle aoe is equal to angle aod is equal to 90 degrees. Now by SAS congruency rule triangle aoe is congruent to triangle aod. We know corresponding parts of congruent triangles are equal. So angle aoe is equal to angle daoe. We also know that angle aoe plus angle daoe is equal to angle aod and angle aod is equal to 60 degrees. So we can write angle daoe plus angle daoe is equal to 60 degrees. We know angle aoe is equal to angle daoe. So we will substitute angle daoe for angle aoe in this expression and we get this expression. Now this further implies angle daoe multiplied by 2 is equal to 60 degrees. Now dividing both the sides of this expression by 2 we get angle daoe is equal to 30 degrees. Now we get angle daoe is equal to angle aoe is equal to 30 degrees. Now let us consider triangle AFB. We know angle AFB is 90 degrees. AF is perpendicular to BC. AF is the height of the cone ABC and height is always perpendicular to the base. So angle AFB is equal to 90 degrees. Now we can write in right triangle AFB tan 30 degrees is equal to BF upon AF. We know tan theta is equal to perpendicular upon base. With respect to this angle BF is the perpendicular and AF is the base. Now we know height of the cone is equal to 20 centimeters. So AF is equal to 20 centimeters and we know value of tan 30 degrees is equal to 1 upon root 3. Now multiplying both the sides of this expression by 20 we get 20 upon root 3 is equal to BF or we can simply write it as BF is equal to 20 upon root 3 centimeters. Similarly we will consider right triangle AOE. So we can write in right triangle AOE tan 30 degrees is equal to OE upon OA. We know tan theta is equal to perpendicular upon base with respect to this angle. This is the perpendicular and this is the base. In triangle AOE let us name this expression as 1. Now we know OA is equal to OF is equal to half AF and AF is equal to 20 centimeters. So substituting 20 centimeters for AF we get OA is equal to OF is equal to 1 upon 2 multiplied by 20 centimeters which is further equal to OA is equal to OF is equal to 10 centimeters. Now substituting OA is equal to 10 centimeters. In expression 1 we get tan 30 degrees is equal to OE upon 10. Now we know tan 30 degrees is equal to 1 upon root 3. So we get 1 upon root 3 is equal to OE upon 10. Now multiplying both the sides of this expression by 10 we get 10 upon root 3 is equal to OE or we can simply write it as OE is equal to 10 upon root 3 centimeters. Now we know BF is equal to R2 is equal to 20 upon root 3 centimeters and R1 is equal to OE is equal to 10 upon root 3 centimeters. These are the two radii of two circular ends of the given frustum. Now let us find out height of the frustum. So we can write height of the frustum that is H is equal to OF and OF is equal to 10 centimeters. Now from key idea we know volume of the frustum is equal to 1 upon 3 pi H multiplied by R1 square plus R2 square plus R1 R2. Now substituting corresponding values of H, R1 and R2 in this expression we get 1 upon 3 pi multiplied by 10 multiplied by square of 10 upon root 3 plus square of 20 upon root 3 plus 10 upon root 3 multiplied by 20 upon root 3 centimeter cube. Now this is further equal to 10 upon 3 pi multiplied by 100 upon 3 plus 400 upon 3 plus 200 upon 3 centimeter cube. Now adding these three terms we get 700 upon 3. So we can write 10 upon 3 pi multiplied by 700 upon 3 centimeter cube. Now this is further equal to 7000 upon 9 pi centimeter cube. So we get volume of the frustum is equal to 7000 pi upon 9 centimeter cube. Now we know frustum is drawn into a wire whose diameter is 1 upon 16 centimeters. So we can write diameter of wire is equal to 1 upon 16 centimeters. Now this implies radius of the wire that is R is equal to 1 upon 32 centimeters. We know radius is equal to half of diameter. So half multiplied by 1 upon 16 is equal to 1 upon 32 centimeters. Now let the length of the wire be L. Now we know volume of cylinder is equal to pi R square H and wire is in the shape of a cylinder. So volume of wire is equal to pi R square L. Here length of the wire is same as height of the cylinder. Now we will substitute corresponding value of radius in this formula and we get pi multiplied by square of 1 upon 32 multiplied by L centimeter cube. Now we know frustum has been drawn into wire. So volume of frustum must be equal to volume of wire. So we can write volume of wire is equal to volume of frustum. Now we know volume of wire is equal to pi multiplied by square of 1 upon 32 multiplied by L and volume of frustum is equal to 7000 pi upon 9. Now pi and pi will get cancelled and we get L is equal to 7000 multiplied by 32 multiplied by 32 upon 9. Now multiplying these three terms we get 716800 upon 9 centimeters is equal to L. Now converting this term to decimals we get L is equal to 796444.44 centimeters. Now this is further equal to 796444.44 upon 100 meters. Now we get length of the wire is equal to 79644444 meters. Now rounding of this value up to one place of decimal we get length of the wire is equal to 796444 meters. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.