Obviously, we're not going to get 24 boxes in there. Our goal should be to arrange the boxes in such a way that the ratios of corresponding sides have the greatest product
Round down:
17/7 = 2
37/7 = 5
27/7 = 3
17/9 = 1
37/9 = 4
27/9 = 3
17/11 = 1
37/11 = 3
27/11 = 2
7m corresponds to the 17m side, which means that the 9m side corresponds to either the 27m or 37m side and the 11m side would correspond to either the 37m or 27m side
(17/7) * (27/9) * (37/11) => 2 * 3 * 3 => 18
(17/7) * (27/11) * (37/9) => 2 * 2 * 4 => 16
17m corresponds to 9m
(17/9) * (27/7) * (37/11) => 1 * 3 * 3 => 9
(17/9) * (37/7) * (27/11) => 1 * 5 * 2 => 10
17m corresponds to 11m side
(17/11) * (27/7) * (37/9) => 1 * 3 * 4 => 12
(17/11) * (27/9) * (37/7) = 1 * 3 * 5 => 15
So 18 is our best arrangement
7m side of smaller boxes lays parallel to the 17m side of the larger box
9m side of smaller boxes lays parallel to the 27m side of the larger box
11m side of smaller boxes lays parallel to the 37m side of the larger box
Not more than 16. 7*2 = 14 17, and 9*4 = 36 < 37, Thus we will place 8 boxes which will occupy 22 m along perpendicular axis. And above that 8 more boxes along vetical axis matching with 27 m edge. Only 3mX1mX5m space will be left out where not even one single piece of 7X9X11 piece can be accomodated.
Answers & Comments
Verified answer
We can establish an upper bound:
(17 * 37 * 27) / (7 * 9 * 11) =>
(17 * 37 * 3) / (7 * 11) =>
((27 - 10) * (27 + 10) * 3) / 77 =>
((729 - 100) * 3) / 77 =>
629 * 3 / 77 =>
1887 / 77 =>
(1540 + 347) / 77 =>
1540/77 + 347/77 =>
20 + (308 + 39) / 77 =>
20 + 308/77 + 39/77 =>
20 + 4 + 0.5 (roughly) =>
24.5
Obviously, we're not going to get 24 boxes in there. Our goal should be to arrange the boxes in such a way that the ratios of corresponding sides have the greatest product
Round down:
17/7 = 2
37/7 = 5
27/7 = 3
17/9 = 1
37/9 = 4
27/9 = 3
17/11 = 1
37/11 = 3
27/11 = 2
7m corresponds to the 17m side, which means that the 9m side corresponds to either the 27m or 37m side and the 11m side would correspond to either the 37m or 27m side
(17/7) * (27/9) * (37/11) => 2 * 3 * 3 => 18
(17/7) * (27/11) * (37/9) => 2 * 2 * 4 => 16
17m corresponds to 9m
(17/9) * (27/7) * (37/11) => 1 * 3 * 3 => 9
(17/9) * (37/7) * (27/11) => 1 * 5 * 2 => 10
17m corresponds to 11m side
(17/11) * (27/7) * (37/9) => 1 * 3 * 4 => 12
(17/11) * (27/9) * (37/7) = 1 * 3 * 5 => 15
So 18 is our best arrangement
7m side of smaller boxes lays parallel to the 17m side of the larger box
9m side of smaller boxes lays parallel to the 27m side of the larger box
11m side of smaller boxes lays parallel to the 37m side of the larger box
(7 * 9 * 11 * 18) / (17 * 27 * 37) =>
7 * 11 * 18 / (17 * 3 * 37) =>
7 * 11 * 6 / (17 * 37) =>
42 * 11 / 629 =>
462/629 =>
0.73449920508744038155802861685215
73.4499% packing efficiency
Not more than 16. 7*2 = 14 17, and 9*4 = 36 < 37, Thus we will place 8 boxes which will occupy 22 m along perpendicular axis. And above that 8 more boxes along vetical axis matching with 27 m edge. Only 3mX1mX5m space will be left out where not even one single piece of 7X9X11 piece can be accomodated.
17 m × 37 m × 27 m = 16983 m^3 (volume)
7 m x 9 m x 11 m = 693 m^3 (volume)
number of boxes = 16983 /693 =24.5 boxes
= 24 boxes