If the length of the diagonal of a cube is 17.32 cm, then the volume of the cube is :

Question

Accepted Answer

To determine the volume of the cube given that the length of its diagonal is 17.32 cm, we can use the relationship between the side length of a cube and its diagonal. The formula for the diagonal $d$ of a cube with side length $a$ is:

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

Given that the diagonal $d$ is 17.32 cm, we can set up the equation:

$$ 17.32 = a\sqrt{3} $$

To find the side length $a$, we solve for $a$:

$$ a = \frac{17.32}{\sqrt{3}} $$

Calculating this gives:

$$ a \approx \frac{17.32}{1.732} \approx 10 \, \text{cm} $$

Now, the volume $V$ of a cube is given by:

$$ V = a^3 $$

Substituting the side length $a = 10 \, \text{cm}$ into the volume formula:

$$ V = 10^3 = 1000 \, \text{cm}^3 $$

Thus, the volume of the cube is 1000 cm³, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

1. **Option 1 (10 cm³):** This option is incorrect because a side length of 10 cm results in a volume of 1000 cm³, not 10 cm³. A volume of 10 cm³ would imply a much smaller side length, which does not match the given diagonal.

2. **Option 2 (100 cm³):** This option is incorrect because a volume of 100 cm³ would imply a side length of approximately 4.64 cm (since $4.64^3 \approx 100$), which would not result in a diagonal of 17.32 cm.

3. **Option 4 (10000 cm³):** This option is incorrect because a volume of 10000 cm³ would imply a side length of 21.54 cm (since $21.54^3 \approx 10000$), which would result in a much larger diagonal than 17.32 cm.

Therefore, the correct answer is Option 3, with a volume of 1000 cm³.