Part 3: The Space Inside! 1. Find the volume of the shipping box using the two methods and show your work: 2. Using the volume formula

9514 1404 393

  36 9/16 cubic feet

Step-by-step explanation:

Volume formula

  V = LWH

  V = (3 3/4 ft)(3 ft)(3 1/4 ft) = (15/4)(3)(13/4) ft³ = 585/16 ft³

  V = 36 9/16 ft³ . . . the volume of the shipping box

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Packing cubes

Each cube measures 1/4 ft on a side. In terms of cubes, the dimensions of the box are ...

  3 3/4 ft = 15/4 ft = 15×(1/4 ft)   ⇒   15 cubes

  3 ft = 12/4 ft = 12×(1/4 ft)   ⇒   12 cubes

  3 1/4 ft = 13/4 ft = 13×(1/4 ft)   ⇒   13 cubes

This means 15 cubes can be lined up along the bottom front of the box. 12 such lines can make one layer of cubes covering the bottom of the box, and 13 such layers will fill the box.

The total number of cubes in the box is ...

  15 × 12× 13 = 2340 . . . . fish food cubes

Each cube has a volume of (1/4 ft)³ = 1/64 ft³, so the volume of the shipping box is ...

  (2340 cubes)×(1/64 ft³/cube) = 2340/64 ft³

  = 36 9/16 ft³ . . . shipping box volume

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Using the volume formula, the volume is 36 9/16 ft³

Using the packing cubes method, the volume is 36 9/16 ft³

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If you consider the math used in the packing cubes method, you see it looks like ...

  V = (15)(12)(13) × (1/64 ft³)

  = (15)(12)(13)×(1/4 ft)³ = (15×1/4 ft)(12×1/4 ft)(13×1/4 ft)

  = (3 3/4 ft)(3 ft)(3 1/4 ft)

  = LWH

That is, the "packing cubes method" is simply a rearrangement of the volume formula product using the commutative and associative properties of multiplication. The same numbers are used to compute the product, but in a different order. Hence the result must be the same.