Question:
If $ \ell $ is the length of a diagonal of a cube of volume $ V $, then:
If $ \ell $ is the length of a diagonal of a cube of volume $ V $, then:
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The diagonal of a cube relates to its side length by $ \ell = a\sqrt{3} $. Using the relationship between the side length and volume ($ a = \sqrt[3]{V} $), you can derive the equation $ 3\sqrt{3} V = \ell^3 $.
Updated On: Jun 5, 2025
- $ 3V = \ell^3 $
- $ \sqrt{3} V = \ell^3 $
- $ 3\sqrt{3} V = 2\ell^3 $
- $ 3\sqrt{3} V = \ell^3 $
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The Correct Option is D
Solution and Explanation
Step 1: Relate the Diagonal of the Cube to Its Side Length.
Let the side length of the cube be \( a \). The volume of the cube is: \[ V = a^3. \] The length of the space diagonal of the cube is given by: \[ \ell = a\sqrt{3}. \] Step 2: Express \( \ell \) in Terms of \( V \).
From \( V = a^3 \), we have: \[ a = \sqrt[3]{V}. \] Substitute \( a = \sqrt[3]{V} \) into the expression for \( \ell \): \[ \ell = \sqrt[3]{V} \cdot \sqrt{3}. \] Cube both sides to eliminate the cube root: \[ \ell^3 = \left(\sqrt[3]{V} \cdot \sqrt{3}\right)^3 = \left(\sqrt[3]{V}\right)^3 \cdot \left(\sqrt{3}\right)^3 = V \cdot 3\sqrt{3}. \] Thus: \[ 3\sqrt{3} \, V = \ell^3. \] Step 3: Analyze the Options.
Option (1): \( 3V = \ell^3 \) — Incorrect, as it does not match the derived equation.
Option (2): \( \sqrt{3} V = \ell^3 \) — Incorrect, as it does not match the derived equation.
Option (3): \( 3\sqrt{3} V = 2\ell^3 \) — Incorrect, as it does not match the derived equation.
Option (4): \( 3\sqrt{3} V = \ell^3 \) — Correct, as it matches the derived equation. Step 4: Final Answer. \[ (4) \mathbf{3\sqrt{3} V = \ell^3} \]
Let the side length of the cube be \( a \). The volume of the cube is: \[ V = a^3. \] The length of the space diagonal of the cube is given by: \[ \ell = a\sqrt{3}. \] Step 2: Express \( \ell \) in Terms of \( V \).
From \( V = a^3 \), we have: \[ a = \sqrt[3]{V}. \] Substitute \( a = \sqrt[3]{V} \) into the expression for \( \ell \): \[ \ell = \sqrt[3]{V} \cdot \sqrt{3}. \] Cube both sides to eliminate the cube root: \[ \ell^3 = \left(\sqrt[3]{V} \cdot \sqrt{3}\right)^3 = \left(\sqrt[3]{V}\right)^3 \cdot \left(\sqrt{3}\right)^3 = V \cdot 3\sqrt{3}. \] Thus: \[ 3\sqrt{3} \, V = \ell^3. \] Step 3: Analyze the Options.
Option (1): \( 3V = \ell^3 \) — Incorrect, as it does not match the derived equation.
Option (2): \( \sqrt{3} V = \ell^3 \) — Incorrect, as it does not match the derived equation.
Option (3): \( 3\sqrt{3} V = 2\ell^3 \) — Incorrect, as it does not match the derived equation.
Option (4): \( 3\sqrt{3} V = \ell^3 \) — Correct, as it matches the derived equation. Step 4: Final Answer. \[ (4) \mathbf{3\sqrt{3} V = \ell^3} \]
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