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The cube and the circles question on unit 0 day 6

Module 0 Day 6 Challenge Part 4
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    mirthfulostrich M0 M1 M2
    last edited by debbie Apr 26, 2020, 9:37 PM Apr 23, 2020, 2:42 PM

    Module 0 Week 2 Day 6 Challenge Part 2

    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.

    D 1 Reply Last reply Apr 25, 2020, 8:11 PM Reply Quote 0
    • D
      debbie ADMIN M0★ M1 M5 @mirthfulostrich
      last edited by debbie Apr 25, 2020, 8:13 PM Apr 25, 2020, 8:11 PM

      @mirthfulostrich Hi, thanks for asking! Maybe let's try thinking backwards, and pretend that we are the little ball with diameter of length 5. From his perspective, he wants to get bigger...

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

      He would like to grow to be like a ball with diameter of 10,10,10, 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 2×2×2=8 2 \times 2 \times 2 = 82×2×2=8 balls in it.

      Similarly, a large ball with twice the dimensions will have 2×2×2=82 \times 2 \times 2 = 82×2×2=8 times the volume of the ball with diameter 5.5.5.

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

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

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

      volume of small ball=12×2×2× volume of the large ball \text{volume of small ball} = \frac{1}{2 \times 2 \times 2} \times \text{ volume of the large ball} volume of small ball=2×2×21​× volume of the large ball

      volume of small ball=12×12×12× 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} volume of small ball=21​×21​×21​× volume of the large ball
      volume of small ball=0.5×0.5×0.5× 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}} volume of small ball=0.5×0.5×0.5× volume of the large ball​

      🙂

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