Given a cube with total surface area S, express its volume V in terms of S.
Answer: Side a = √(S6), so V = (S6)3/2 = S6√(S6).
Step-by-step solution
Idea: Find the side from the surface area (S = 6a2), then cube it.
- S = 6a2 ⇒ a2 = S6 ⇒ a = √(S6).1 mark
- V = a3 = (S6)3/2 = S6√(S6) (also = S√S6√6).1 mark
V = (S/6)^(3/2) = (S/6)√(S/6)
Check: S = 150: a = 5, V = 125; (1506)3/2 = 253/2 = 125 ✓.
Answer to write in the exam
a2 = S6 ⇒ a = √(S6)
V = a3
∴ V = (S6)3/2
Common mistakes that cost marks
- Writing V = (S6)3, forgetting that S6 is a2, not a.
- Writing V = S6 × a without replacing a.
How this can come in the exam
MCQ (1 mark)
For a cube of total surface area S and volume V, V2 equals
- S3216
- S336
- S236
- S36
Show answer
(A) S3216
a2 = S6, so V2 = a6 = (S6)3 = S3216. (Check: a = 2 gives S = 24, V = 8, and 243 ÷ 216 = 64 = 82.)
Try one yourself
Find V when S = 864 cm2.
Show answer
8646 = 144 ⇒ a = 12 ⇒ V = 1728 cm3.
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