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 m^{3}.

Determine the dimensions of Stefan and Lali’s boxes.

Let the dimensions of the boxes be:

Length(L) = 4k

Width(W) = 3k

Height(H) = 5k

Let L+W+H = P ( say). Now, its given that:

100< P < 1000

=> 100 < 12k < 1000

=> 8 +1/3 < k < 83 + 1/3

=> 9 <= k <= 83

Thus, the respective minimum and maximum value of k are 9 and 83.

Hence the dimension of the box bought by Stefan

= 36 x 27 x45 and its volume :

= 36*27*45 cm^3

= 43740 cm^3

= 0.4374 m^3

Now, the dimensions of the box bought by Lali is contingent upon its volume which is < 4 m^3

Therefore, we must have:

60*k^3< 4,000,000 cm^3

=> k^3 < 66666+2/3

=> k < 40.54

Then substituting k = 40, we have the dimensions of the box bought by Lali

= 160 x 120 x 200

And its volume

= 60 *40^3

= 3840000

= 3.84 m^3 < 4 m^3

Consequently, the respective dimensions of the two boxes bought by Stefan and Lali are 36 x 27 x 45 and: 160 x120 x 200.

*Edited on ***January 30, 2024, 12:31 pm**