For the part when you talk about how the small cube times 3x3x3 is equal to the cube that is 27 units I understand however I don't understand why the ball with the diameter of 10 multiplies by 0.5x0.5x0.5 to get to the ball with the diameter of 5.

]]>He would like to grow to be like a ball with diameter of \(10,\) which means he wants to double in all three dimensions:

From looking at this, his volume definitely doesn't double.... that would give a snowman. We don't want a snowman!

Another illustration of why his volume doesn't double. We don't want a caterpillar!

We want something like a ball, with twice the width, twice the length, and twice the height. In terms of balls, it would be like this:

The stack of balls has \( 2 \times 2 \times 2 = 8\) balls in it.

Similarly, a large ball with twice the dimensions will have \(2 \times 2 \times 2 = 8\) times the volume of the ball with diameter \(5.\)

$$ \text{volume of large ball} = 2 \times 2 \times 2 \times \text{ volume of the small ball} $$

Inverting this to solve for the volume of the small ball, we get

$$ \text{volume of small ball} = \frac{1}{2 \times 2 \times 2} \times \text{ volume of the large ball} $$

$$ \text{volume of small ball} = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \text{ volume of the large ball} $$

$$ \boxed{\text{volume of small ball} = 0.5 \times 0.5 \times 0.5 \times \text{ volume of the large ball}} $$