Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.
- (i) 6a2 − 24b2
- (ii) 3ps2 − 15ps + 12p
Step-by-step solution
Idea: Volume of a cuboid = length × breadth × height, so we need to write the volume as a product of three factors. First take out the common factor, then factorise what is left using an identity. The three factors are possible expressions for the three dimensions.
(i) 6a2 − 24b2
- 6 divides both 6 and 24, so take 6 out as a common factor: 6a2 − 24b2 = 6(a2 − 4b2).½ mark
- a2 − 4b2 = a2 − (2b)2 is a difference of two squares, so use x2 − y2 = (x − y)(x + y): a2 − 4b2 = (a − 2b)(a + 2b).½ mark
- So volume = 6 × (a − 2b) × (a + 2b). Possible dimensions: 6 units, (a − 2b) units and (a + 2b) units (this needs a > 2b, so that every length is positive).½ mark
- Other splits are also possible, because the 6 can be shared out between the factors, e.g. 2, 3(a − 2b) and (a + 2b), or 3, (2a − 4b) and (a + 2b).
(ii) 3ps2 − 15ps + 12p
- Every term contains 3 and p, so take out 3p: 3ps2 − 15ps + 12p = 3p(s2 − 5s + 4).½ mark
- s2 − 5s + 4 has the shape s2 + (a + b)s + ab, so use (s + a)(s + b) = s2 + (a + b)s + ab. We need two numbers that add to −5 and multiply to 4: they are −1 and −4. So s2 − 5s + 4 = (s − 1)(s − 4).½ mark
- So volume = 3p × (s − 1) × (s − 4). Possible dimensions: 3p units, (s − 1) units and (s − 4) units (this needs p > 0 and s > 4).½ mark
- Other splits are possible too, e.g. 3, p(s − 1) and (s − 4), or p, (3s − 3) and (s − 4).
Check: (i) a = 3, b = 1: volume 54 − 24 = 30 and 6 × 1 × 5 = 30 ✓. (ii) p = 2, s = 6: volume 216 − 180 + 24 = 60 and 6 × 5 × 2 = 60 ✓.
Answer to write in the exam
(i)
Volume = 6a2 − 24b2 = 6(a2 − 4b2)
= 6[a2 − (2b)2]
= 6(a − 2b)(a + 2b) [a2 − b2 = (a + b)(a − b)]
∴ Length = 6 units, breadth = (a − 2b) units, height = (a + 2b) units
(ii)
Volume = 3ps2 − 15ps + 12p = 3p(s2 − 5s + 4)
= 3p(s2 − s − 4s + 4)
= 3p[s(s − 1) − 4(s − 1)]
= 3p(s − 1)(s − 4)
∴ Length = 3p units, breadth = (s − 1) units, height = (s − 4) units
Common mistakes that cost marks
- Stopping after taking out the common factor, e.g. giving only two dimensions 6 and (a2 − 4b2). A cuboid needs three dimensions, so keep factorising.
- Choosing +1 and +4 for s2 − 5s + 4. They multiply to 4 but add to +5, not −5. The correct pair is −1 and −4.
- Forgetting the p when taking out the common factor in (ii), and writing 3(s − 1)(s − 4), which is a different volume.
How this can come in the exam
The volume of a cuboid is (2x3 − 8x) cubic units. Possible dimensions are
- 2x, (x − 2), (x + 2)
- 2, (x − 4), (x + 4)
- 2x, (x − 4), (x + 4)
- x, (x − 2), (x + 2)
Show answer
(A) 2x, (x − 2), (x + 2)
2x3 − 8x = 2x(x2 − 4) = 2x(x − 2)(x + 2).
The volume of a cuboid is (x3 − 5x2 + 6x) cubic units. Find possible expressions for its dimensions.
Show answer
Take out x: x(x2 − 5x + 6) (½ mark). Two numbers adding to −5 and multiplying to 6: −2 and −3 (½ mark). So volume = x(x − 2)(x − 3), and the dimensions can be x, (x − 2) and (x − 3) units (1 mark).Try one yourself
The volume of a cuboid is (2m2n − 18n) cubic units. Find possible expressions for its length, breadth and height.
Show answer
2m2n − 18n = 2n(m2 − 9) = 2n(m − 3)(m + 3). Dimensions: 2n, (m − 3), (m + 3) units.
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