What is the volume (in cm3) of a cube whose space diagonal is \(4\sqrt{3}\) cm?
2023
What is the volume (in cm3) of a cube whose space diagonal is \(4\sqrt{3}\) cm?
Answer: B. 64 — CONCEPTFor a cube with edge length a, the space diagonal is d = a√3 and the volume is V = a3. Thus a known positive space diagonal determines one edge length…
- A.
8
- B.
64
- C.
16
- D.
More than one of the above
- E.
None of the above
Show answer & explanation
Correct answer: B
CONCEPT
For a cube with edge length a, the space diagonal is d = a√3 and the volume is V = a3.
Thus a known positive space diagonal determines one edge length and therefore one volume.
APPLICATION
Let the edge length be a cm.
Use the space-diagonal relation d = a√3.
Substitute d = 4√3: a√3 = 4√3.
Divide by √3 to obtain a = 4 cm.
Now V = a3 = 43 = 64 cm3.
CROSS-CHECK
With a = 4 cm, the diagonal is √(42 + 42 + 42) = √48 = 4√3 cm, which reproduces the given diagonal. Therefore, the volume is 64 cm3.