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If the length of the diagonal of a cube is [tex]4 \sqrt{3} \, \text{cm}[/tex], then the volume of the cube is:

A. [tex]64 \, \text{cm}^3[/tex]

B. [tex]48 \, \text{cm}^3[/tex]

C. [tex]16 \, \text{cm}^3[/tex]

D. [tex]4 \, \text{cm}^3[/tex]


Sagot :

To find the volume of a cube when given the length of its diagonal, let's go through the steps systematically.

1. Understand the Relationship:
- For a cube, the length of the diagonal (d) can be found using the formula:
[tex]\[ d = a\sqrt{3} \][/tex]
where [tex]\( a \)[/tex] is the side length of the cube.

2. Given:
- The length of the diagonal of the cube is [tex]\( 4\sqrt{3} \)[/tex] cm.

3. Find the Side Length:
- We need to solve for [tex]\( a \)[/tex]:
[tex]\[ 4\sqrt{3} = a\sqrt{3} \][/tex]
- Divide both sides by [tex]\( \sqrt{3} \)[/tex]:
[tex]\[ a = \frac{4\sqrt{3}}{\sqrt{3}} = 4 \text{ cm} \][/tex]

4. Calculate the Volume:
- The volume [tex]\( V \)[/tex] of a cube is given by:
[tex]\[ V = a^3 \][/tex]
- Substitute [tex]\( a = 4 \text{ cm} \)[/tex]:
[tex]\[ V = 4^3 = 64 \text{ cm}^3 \][/tex]

Therefore, the volume of the cube is [tex]\( 64 \text{ cm}^3 \)[/tex].
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