M0W2D6-ball-diameter-5-to-10-forum.png

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

M0W2D6-ball-diameter-5-to-10-forum2.png

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

M0W2D6-ball-diameter-5-to-10-forum3.png

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

M0W2D6-ball-diameter-5-to-10-forum5.png

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:

M0W2D6-ball-diameter-5-to-10-forum6.png

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 \(5.\)

M0W2D6-ball-diameter-5-to-10-forum7.png

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

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

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

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

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

🙂

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