Suppose for the moment that the inequality holds for all :
Expanding everything gives
In particular, the inequality says that for any choice of
. But if
and
, then
while
.
So one possible choice of could be
and
. Then we get
but clearly it's not true that .
we have 630 one-inch unit cubes and we want to completely fill the rectangular box (unknown dimensions).
If all the cubes are fitted tightly inside rectangular box without living any space, then box volume would be equal to cubes volume.
There are 630 one-inch unit cubes, so volume of cubes = 630 cubic inches.
Now the volume of rectangular box would also be 630 cubic inches.
We know the formula for volume of rectangular box = length × width × height.
So we need to find any three positive integers whose product is 630.
Out of all given choices, only option A satisfies the condition of factors of 630.
Hence, option A i.e. (7 in x 9 in x 10 in) is the final answer.
