A solid metallic cube of side 12 cm is melted and recast into solid cylindrical rods, each having radius 2 cm and height 12 cm. Find:
- (i) the volume of the cube,
- (ii) the volume of one cylindrical rod,
- (iii) the approximate number of complete cylindrical rods that can be formed. (π ≈ 227)
Step-by-step solution
Idea: Melting and recasting keeps the volume of metal the same. Number of rods = volume of cube ÷ volume of one rod; only whole rods count.
(i) the volume of the cube,
- Volume = 123 = 12 × 12 × 12 = 1728 cm3.½ mark
(ii) the volume of one cylindrical rod,
- Volume = πr2h = 227 × 22 × 12 = 22 × 487.½ mark
- = 10567 ≈ 150.86 cm3.½ mark
(iii) the approximate number of complete cylindrical rods that can be formed. (π ≈ 227)
- Number = 17281056/7 = 1728 × 71056 = 120961056 ≈ 11.45.1 mark
- Only complete rods count, so 11 rods (some metal is left over).½ mark
Check: 11 rods use 11 × 150.86 ≈ 1659.4 cm3 ≤ 1728 ✓; 12 rods would need 1810.3 cm3 > 1728 ✗. So 11 ✓.
Answer to write in the exam
(i)
Volume of cube = 123
∴ Volume = 1728 cm3
(ii)
Volume of rod = πr2h = 227 × 4 × 12
∴ Volume = 10567 ≈ 150.86 cm3
(iii)
Number of rods = 1728150.86 ≈ 11.45
∴ 11 complete rods can be formed
Common mistakes that cost marks
- Rounding 11.45 up to 12. A 12th rod cannot be completed.
- Using the diameter (4 cm) as the radius.
- Thinking surface area is conserved on melting. Volume is conserved, not surface area.
How this can come in the exam
A metal cuboid 22 cm × 14 cm × 10 cm is melted into cylindrical coins of radius 1 cm and thickness 0.2 cm. How many coins are made? (π = 227)
- 2450
- 4900
- 3080
- 4400
Show answer
(B) 4900
Cuboid 3080 cm3; coin 227 × 1 × 0.2 = 4.47 cm3; 3080 × 74.4 = 4900.
Try one yourself
A cube of side 10 cm is melted into rods of radius 1 cm and length 7 cm. How many complete rods are made? (π = 227)
Show answer
Rod = 227 × 1 × 7 = 22 cm3; 1000 ÷ 22 ≈ 45.45, so 45 rods.
More questions like this
- What is the curved surface area of a cone?
- What is the volume of a cone?
- Here is another hands-on activity that uses modelling clay (earlier this used to be called ‘plasticine’). Carefully mold the clay into a solid cylinder and measure its radius and height. Then reshape the same portion of clay into identical cones, each having the same radius and the same height as the cylinder. You will find that you are able to make exactly three such cones.
- A right triangle ABC with sides 5 cm, 12 cm and 13 cm is rotated through 360° about the side with length 12 cm. Find the volume of the solid so obtained.
- Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.