The radius of the base of a cylinder is increased by 10%. At the same time, the height of the cylinder is decreased by x%. Given that the volume of the cylinder remains unchanged, find the value of x.
Step-by-step solution
Idea: New radius = 1.1r. For the volume πr2h to stay the same, the height must shrink by the factor 11.21.
- New radius = 1.1r; new height = h(1 − x100).½ mark
- Volume unchanged: π(1.1r)2 × h(1 − x100) = πr2h ⇒ 1.21(1 − x100) = 1.1 mark
- 1 − x100 = 11.21 = 100121, so x100 = 21121.1 mark
- x = 2100121 = 1743121 ≈ 17.36.½ mark
Check: Take r = 10, h = 121: volume = π × 100 × 121. New: radius 11, height 121 × (1 − 0.1736) = 100: π × 121 × 100 ✓, the same.
Answer to write in the exam
π(1.1r)2 × h(1 − x100) = πr2h
1.21(1 − x100) = 1
1 − x100 = 100121 ⇒ x100 = 21121
∴ x = 2100121 ≈ 17.36
Common mistakes that cost marks
- Answering 10% or 20% by thinking ‘radius up 10%, so height down 10%’ or ‘squared, so 20%’. The exact factor is 11.21.
- Writing (1.1)2 = 1.21 as 1.12.
- Writing x = 21 from 1.21 − 1 = 0.21, which is the increase in r2, not the decrease in height.
How this can come in the exam
The radius of a cylinder is doubled. To keep the volume the same, the height must be
- halved
- divided by 4
- doubled
- unchanged
Show answer
(B) divided by 4
Radius × 2 makes r2 × 4, so height ÷ 4.
The radius of a cylinder is decreased by 20%. By what percentage must its height be increased so that the volume is unchanged?
Show answer
(0.8)2(1 + y100) = 1 ⇒ 1 + y100 = 10.64 = 1.5625 ⇒ y = 56.25%.Try one yourself
The height of a cylinder is increased by 44%. By what percentage must the radius be decreased to keep the volume the same?
Show answer
r‘2 × 1.44 = r2 ⇒ r‘ = r1.2 ≈ 0.8333r: decrease of about 16.67% (exactly 1623%).
More questions like this
- 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:
- 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.