Box Volume Solution
Volume equals length times depth times height, V=ldh. We see in the table and graph below that maximum volume for our Lego storage is at a height of 2 inches, length and depth 8 inches.
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Two and eight are related: 8 is the third power of 2. What we see here is 2 inches on either end of an 8-inch side, and this combination is the optimal ratio of powers.
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When we have 1-inch height and 10-inch sides or 5-inch height and 2-inch sides, we don't have any clear relationship, so these values being sub-optimal make sense.
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The scenario with 3-inch height and 6-inch sides has a multiplicative relationship but not an exponential one.
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The remaining scenario, 4-inch height and 4-inch sides, yields 4-cubed, or 2 to the sixth. Comparatively, 8-squared times 2 yields 2-cubed times 2-cubed times 2, or 2 to the seventh.