The volume of a cube, whose diagonal is 2.5 meter, is equal to :

Question

Accepted Answer

To determine the volume of a cube whose diagonal is 2.5 meters, we need to use the relationship between the side length of the cube and its diagonal. The diagonal $d$ of a cube with side length $a$ is given by the formula:

$$ d = a\sqrt{3} $$

Given that the diagonal $d$ is 2.5 meters, we can set up the equation:

$$ 2.5 = a\sqrt{3} $$

Solving for $a$, we divide both sides by $\sqrt{3}$:

$$ a = \frac{2.5}{\sqrt{3}} $$

To find the volume $V$ of the cube, we use the formula for the volume of a cube:

$$ V = a^3 $$

Substituting the expression for $a$, we have:

$$ V = \left(\frac{2.5}{\sqrt{3}}\right)^3 $$

Calculating this, we get:

$$ V = \frac{2.5^3}{(\sqrt{3})^3} $$

$$ V = \frac{15.625}{3\sqrt{3}} $$

To simplify further, we rationalize the denominator:

$$ V = \frac{15.625 \times \sqrt{3}}{3 \times 3} $$

$$ V = \frac{15.625\sqrt{3}}{9} $$

Now, let's convert 15.625 to a fraction:

$$ 15.625 = \frac{125}{8} $$

Substituting this into the volume expression:

$$ V = \frac{\frac{125}{8} \times \sqrt{3}}{9} $$

$$ V = \frac{125\sqrt{3}}{72} $$

This matches Option 1: $\frac{125\sqrt{3}}{72}$.

Now, let's explain why the other options are incorrect:

1. **Option 2: $\frac{625}{8}$** - This option does not involve $\sqrt{3}$, which is essential in the calculation due to the diagonal formula $d = a\sqrt{3}$. Therefore, it cannot be correct.

2. **Option 3: $\frac{125\sqrt{2}}{32}$** - This option incorrectly uses $\sqrt{2}$ instead of $\sqrt{3}$. The diagonal of a cube involves $\sqrt{3}$, not $\sqrt{2}$, making this option incorrect.

3. **Option 4: $\frac{128}{7}$** - This option neither involves $\sqrt{3}$ nor matches the calculated volume. It does not align with the necessary calculations for the cube's volume based on its diagonal.

Thus, Option 1 is the correct answer, as it accurately represents the volume of the cube given the diagonal of 2.5 meters.