Problems Plus 1: Tightly Packed Balls in a Box (and Tightly Packed Circles in a Square)

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In this video I determine the radius of 9 balls that are tightly packed inside a box. I first solve the 2D case by solving for the radius of 5 circles tightly packed inside a square. The strategy involves first determining the diagonal, using the Pythagorean Theorem in 2D and 3D, and then showing that the diagonal is also equal to 2r + 4x, where r is the radius and x is the distance from the center to the corner. From here we can use basic algebra to solve for the radius.

Timestamps:

• Problem 1: Tightly packed balls in a box: 0:00
• Hint: Use analogy problem solving strategy: 1:06
• Solution: Solving the 2D case first: 2:25
  • Diagonal of a square is 2^(1/2): 4:11
  • Diagonal = 4r and 2 x: 4:59
  • Radius solved for the 2D case: 6:56
• Solving the 3D case: Diagonal is 3^(1/3): 7:07
  • Drawing our balls in 3D: 9:14
  • Diagonal = 4r + 2x: 11:48
  • Radius = 3^(1/2) - 3/2: 15:27

Video notes and playlists:

• HIVE notes: https://peakd.com/hive-128780/@mes/jlgigkdl
• Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0G_7Jd6ZORHFQMk8jbwmkos
• Vectors and the Geometry of Space playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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