Question

The volume of a cubical box is 474.552 cubic metres. Find the length of each side of the box.

Answer 1

Volume of a cube is given by:

\[V = s^3\], where s = side of the cube

Now

\[s^3 = 474 . 552 \text{ cubic metres } \]

\[ \Rightarrow s = \sqrt[3]{474 . 552} = \sqrt[3]{\frac{474552}{1000}} = \frac{\sqrt[3]{474552}}{\sqrt[3]{1000}}\]

To find the cube root of 474552, we need to proceed as follows:
On factorising 474552 into prime factors, we get:

\[474552 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 13 \times 13 \times 13\]

On grouping the factors in triples of equal factors, we get:

\[474552 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 13 \times 13 \times 13 \right\}\]

Now, taking one factor from each triple, we get:

\[\sqrt[3]{474552} = \sqrt[3]{\left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 13 \times 13 \times 13 \right\}} = 2 \times 3 \times 13 = 78\]

Also

\[\sqrt[3]{1000} = 10\]

∴ \[s = \frac{\sqrt[3]{474552}}{\sqrt[3]{1000}} = \frac{78}{10} = 7 . 8\]

Thus, the length of the side is 7.8 m.