### Video Transcript

Consider a cube of side two 𝑥 centimeters. 𝑥 is an integer. The total volume of the cube is greater than 1111 centimeters cubed. Show that 𝑥 is greater than or equal to six.

The volume of a cube can be calculated by cubing the length of one of the sides. In our example, as the side is two 𝑥 centimeters, we need to cube two 𝑥. This is equal to eight 𝑥 cubed.

As the total volume of the cube is greater than 1111, we can write this as an inequality: eight 𝑥 cubed is greater than 1111. Dividing both sides of the inequality by eight gives us 𝑥 cubed is greater than 138.875. Cube rooting both sides of this inequality gives us 𝑥 is greater than 5.1785 to four decimal places.

We were told in the question that 𝑥 is an integer. This means that it must be a whole number. Since 𝑥 is an integer, 𝑥 must be greater than or equal to six. If 𝑥 is equal to six, seven, eight, and so on, then the total volume of the cube will be greater than 1111 centimeters cubed.

As we know that 𝑥 is an integer, an alternative method would be to substitute 𝑥 equals six and 𝑥 equals five into the inequality eight 𝑥 cubed is greater than 1111. Eight multiplied by six cubed is equal to 1728 and 1728 is greater than 1111. Eight multiplied by five cubed is equal to 1000 and 1000 is less than 1111.

This means that when 𝑥 is equal to five, the volume will be less than 1111. But when 𝑥 is equal to six, it will be greater than 1111. This method leads us to the same conclusion that since 𝑥 is an integer, 𝑥 must be greater than or equal to six.