Name: Problems Plus 1: Tightly Packed Balls in a Box (and Tightly Packed Circles in a Square)
Uploaded: 2024-10-03T04:22:10+00:00
Duration: 15 min 57 s
Description: 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 invo

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Health & Science Education math calculus 3D lines vector geometry STEM science tightly packed tight packing

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