Given below are two quantities named A and B. Based on the given information, you have to determine relation between the two quantities. You should use the given data and knowledge of Mathematics to choose between the possible answers. Quantity A: Number of hemispheres that can be formed by melting a sphere having volume equal to the volume of right circular cylinder whose diameteris 6 cm and height is 4 cm. Also, radius of hemisphere formed is half the radius of sphere from which the hemisphere is formed. Quantity B: Number of ice cream cones that can be made from a right circular cylinder shape ice cream container having radius 6 cm and height 5 cm. Ice cream cone to be made should have height 4 cm and slant height 5 cm and there should be a hemisphere made on top of the cone having radius equal to the radius of cone.?

Option 1 : Quantity A > Quantity B

Quantity A:

Let the number of hemispheres that can be formed = x

Radius of hemisphere to be formed = R_h

⇒ Volume of cylinder = π × r² × h = π × (6/2)² × 4 = π × 3² × 4 = 36π

Now,

 Volume of sphere = volume of cylinder

⇒ (4/3) × π × R³ = 36 × π

⇒ R = 3 cm

⇒ Radius of hemisphere forms = R_h = R/2 = 3/2

Now, since hemisphere is formed by melting sphere,

⇒ Volume of all the hemisphere formed = volume of sphere

⇒ x × {(2/3) × π × R_h ³} = 36 × π

⇒ x × {(2/3) × π × (3/2) ³} = 36 × π

⇒ x × {(2/3) × π × (27/8)} = 36 × π

⇒ x = 16

So, 16 hemispheres can be formed from melting the sphere

⇒ Quantity A = 16

Quantity B:

Let number of ice cream cones that can formed = x

Now, Ice cream container is a cylinder having radius 6 cm and height 5 cm,

⇒ Total volume of ice cream in container = volume of cylinder = π × r² × h = π × 6² × 5 = 180 × π

Now, it is given that ice cream formed should also have a hemisphere built on it,

⇒ Total volume of 1 ice cream cone = volume of cone + volume of hemisphere

Given,

⇒ height of cone given = h_c= 4 cm

⇒ Slant height of cone = l = 5 cm

We know, r_c² + h_c² = l²

⇒ r_c² + 16 = 25

⇒ r_c² = 25 – 16 = 9

⇒ r_c = 3

⇒ Volume of cone = (1/3) ×π × r² × h = (1/3) × π × 3² × 4 = 12 × π

⇒ Volume of hemisphere = (2/3) × π × r_h³

As it is given that radius of hemisphere = radius of cone

⇒ r_h = 3

⇒ Volume of hemisphere = (2/3) × π × 3³ = 18 × π

⇒ Total volume of one ice cream cone = (12 × π) + (18 × π) = 30 × π

⇒ Total number of cones = total volume of ice cream/volume of one ice cream cone

⇒ Number of cones = (180 × π)/(30 × π) = 6

⇒ Number of cones that can be made = 6

⇒ Quantity B = 6