Multiple choice

Solid cylinders of equal volume are tightly packed in two layers in a rectangular box such that in each layer there are three rows of four such cylinders Find the percentage of volume of empty space in the box approximately.

  1. $19.27 \%$
  2. $20.54 \%$
  3. $21.43 \%$
  4. $22.36 \%$
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C Correct answer
Explanation

The box contains 2 layers * 3 rows * 4 cylinders = 24 cylinders. Each cylinder has volume V = pi * r^2 * h. The box dimensions are 4 * (2r) by 3 * (2r) by 2 * h. Box volume = 8r * 6r * 2h = 96 * r^2 * h. Total cylinder volume = 24 * pi * r^2 * h. Empty space = 96 - 24 * pi. Percentage = (96 - 24 * 3.14159) / 96 * 100 = (96 - 75.398) / 96 * 100 = 21.46%.

AI explanation

Let the radius of each of the 24 cylinders be r and the height of the rectangular box be 2r to accommodate two layers. Since the cylinders are tightly packed in a 3 by 4 grid per layer, the dimensions of the box are 8r by 6r by 2r, giving a total box volume of 96r^3. The volume of a single cylinder is πr^2h, and substituting h = 2r gives 2πr^3, making the total volume of all 24 cylinders 48πr^3. The percentage of empty space is found by subtracting the cylinders' volume from the box volume, dividing by the box volume, and multiplying by 100, which is ((96r^3 - 48πr^3) / 96r^3) times 100 = (1 - π/2) times 100, yielding approximately 21.43 percent.

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