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Change in volume · 4 marks

What is the change in volume when:

  1. (i) the length of a cuboid with dimensions l, w, h is increased by 1 unit?
    (a) 1 cubic unit (b) (lwh + 1) cubic unit (c) wh cubic units (d) lw cubic units (e) lh cubic units
  2. (ii) the radius of a cylinder with dimensions r, h is increased by 1 unit?
    (a) 1 cubic unit (b) πr2h cubic units (c) πr2 cubic units (d) 2πrh + 2πh cubic units (e) 2πrh + πh cubic units
  3. (iii) the radius of a sphere is decreased by 1 unit?
Answer: (i) (c) wh cubic units (ii) (e) 2πrh + πh cubic units (iii) the volume decreases by 43π(3r2 − 3r + 1) = 4πr2 − 4πr + 43π cubic units

Step-by-step solution

Idea: Change in volume = new volume − old volume. Write the new volume with the changed measurement and subtract.

(i) the length of a cuboid with dimensions l, w, h is increased by 1 unit?
(a) 1 cubic unit (b) (lwh + 1) cubic unit (c) wh cubic units (d) lw cubic units (e) lh cubic units

  1. New volume = (l + 1)wh = lwh + wh. Change = wh: a slab 1 unit thick on the w × h face. Answer (c).1 mark
(c) wh cubic units

(ii) the radius of a cylinder with dimensions r, h is increased by 1 unit?
(a) 1 cubic unit (b) πr2h cubic units (c) πr2 cubic units (d) 2πrh + 2πh cubic units (e) 2πrh + πh cubic units

  1. New volume = π(r + 1)2h = π(r2 + 2r + 1)h.½ mark
  2. Change = π(2r + 1)h = 2πrh + πh. Answer (e).1 mark
(e) 2πrh + πh cubic units

(iii) the radius of a sphere is decreased by 1 unit?

  1. Change = 43πr3 − 43π(r − 1)3 = 43π[r3 − (r3 − 3r2 + 3r − 1)].1 mark
  2. = 43π(3r2 − 3r + 1) = 4πr2 − 4πr + 43π. The volume decreases by this amount.½ mark
Decrease of 43π(3r2 − 3r + 1) cubic units
(i) (c) wh (ii) (e) 2πrh + πh (iii) decreases by (4/3)π(3r² − 3r + 1) = 4πr² − 4πr + 4π/3 cubic units

Check: (ii) r = 1, h = 1: π × 4 − π × 1 = 3π and 2π + π = 3π ✓. (iii) r = 2: 43π(8 − 1) = 283π and 43π(12 − 6 + 1) = 283π ✓.

Answer to write in the exam

(i)

Change = (l + 1)wh − lwh = wh

∴ (c) wh cubic units

(ii)

Change = π(r + 1)2h − πr2h

= π(2r + 1)h = 2πrh + πh

∴ (e)

(iii)

Decrease = 43πr3 − 43π(r − 1)3

= 43π(3r2 − 3r + 1)

∴ Volume decreases by 4πr2 − 4πr + 43π cubic units

Common mistakes that cost marks

  • Choosing (a) 1 cubic unit, thinking adding 1 unit to a side adds 1 cubic unit.
  • In (ii), writing (r + 1)2 = r2 + 1 and choosing (c).
  • In (iii), writing (r − 1)3 = r3 − 1.

How this can come in the exam

MCQ (1 mark)

The side of a cube is increased from a to a + 1. The increase in its surface area is

  1. 6
  2. 12a + 6
  3. 6a + 6
  4. 12a
Show answer

(B) 12a + 6
6(a + 1)2 − 6a2 = 12a + 6.

Try one yourself

By how much does the volume of a cone change when its height increases by 1 unit (radius fixed)?

Show answer

13πr2(h + 1) − 13πr2h = 13πr2 cubic units.

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