Question

If the volume of a cube is \(125\ \text{cm}^3\) then the ratio of the side of the cube and the space diagonal of the cube is:

• A. \(1 : \sqrt{3}\)
• B. \(5 : \sqrt{3}\)
• C. \(25 : \sqrt{3}\)
• D. \(15 : \sqrt{3}\)

Hint:

Cube facts: \(V=a^3\), face diagonal \(=a\sqrt{2}\), space diagonal \(=a\sqrt{3}\). Ratios often simplify by canceling \(a\).

Answer 1

The Correct Option is A

Step 1: Find the side length from volume.
For a cube with side \(a\), volume \(V=a^3\). Given \(a^3=125 \Rightarrow a=\sqrt[3]{125}=5\ \text{cm}.\)
Step 2: Write the space diagonal of a cube.
Space diagonal \(d=a\sqrt{3} = 5\sqrt{3}\ \text{cm}.\)
Step 3: Form the required ratio (side : diagonal).
\[ a:d = 5 : 5\sqrt{3} = 1 : \sqrt{3}. \]