perplexus.info :: Just Math : Picking boxes

Picking boxes (Posted on 2024-01-28)

Billy’s Box Company sells boxes with the following very particular restrictions on their dimensions.

1. The length, width, and height, in cm, must be all integers.
2. The ratio of the length to the width to the height must be 4:3:5.
3. The sum of the length, width, and height must be between 100 cm and 1000 cm, inclusive.

Stefan bought the box with the smallest possible volume, and Lali bought the box with the largest volume less than 4 m³.

Determine the dimensions of Stefan and Lali’s boxes.

No Solution Yet Submitted by Danish Ahmed Khan

Rating: 3.0000 (1 votes)

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Solution

| Comment 3 of 5 |

Sides are (3k,4k,5k)

Sum of 3 is 100 < 12k < 1000; so 8 < k < 84

Vol is 60k^3

Smallest possible box is 60 * 9^3 = 43740 cm^3 = 0.04374 m^3

Largest possible box is 60 * 83^3 = 34307220 cm^3 = 34.30722 m^3

but Lali's bought one less than m^3

60k^3 < 4000000

k^3 < 66666

k < about 40.5

if k = 40

Volume = 60 * 40^3 = 3,840,000 cm^3 = 3.84 m^3

But if k = 41; 60 * 41^3 = 4,135,260 cm^3 > 4 m^3

Stefan's box in cm: 36 x 27 x 45 with volume 0.04374 m^3

Lali's box in cm: 160 x 120 x 200 with volume 3.84 m^3

Posted by Larry on 2024-01-28 07:54:08

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