Question

Calculate the number of different sized rectangular solids with a volume of 32 cubic units such that each dimension has an integer value.
(a) 4
(b) 5
(c) 6
(d) 10

Answer 1

Hint: We solve this problem by using the formula of volume of a cuboid.
We have the formula for the volume of a cuboid having length \[l\], breadth \[b\] and height \[h\] as
\[V=lbh\]
By using the above formula we take all possibilities of different combinations of length, breadth, and height of different dimensions to the given volume.

Complete step by step answer:
We are asked to find the number of cuboids having a volume 32 cubic units.
We know that the formula for the volume of a cuboid having length \[l\], breadth \[b\] and height \[h\] as
\[V=lbh\]
By using the above formula to the given volume we get
\[\Rightarrow lbh=32\]
We are given that the dimensions should be rectangular and are integers
We know that for being a rectangular figure than any two of length, breadth and height may be equal.
Now, let us take all the possibilities of length, breadth, and height such that the product will be 32 by using the given conditions as follows
(1) 1, 1, 32
(2) 1, 2, 16
(3) 1, 4, 8
(4) 2, 2, 8
(5) 2, 4, 4
Here, we can see that there are a total of if 5 sets of numbers.
Therefore, we can conclude that the number of rectangular solids of volume 32 is 5
So, option (b) is the correct answer.


Note:
 Students may do mistakes in considering the different sized rectangular solids.
Here, the different sized doesn’t mean that all the sides are different. There may be a possibility of taking two sides equal so that the set forms a rectangular solid.
Therefore, we get 5 rectangular solids.
But students may miss the possibilities of taking any two sides as equal and give the answer as 2
Also exchanging the dimensions within the set count only one solid that is let us consider the following sets
(1) 1, 2, 16
(2) 2, 1, 16
Here, the above two sets represent the same solid so that we should count them as 1.