Question

If l, b, h of a cuboid increased, decreased and increased by 1%, 3%, 2% respectively then the volume of the cuboid
A) Increases
B) Decreases
C) Increases or decreases depending on the original dimensions
D) Can’t be calculated with given data

Answer 1

Hint: Let us consider the length, breadth and height of a cuboid is \[l,b,h\] respectively.
Then the volume of the cuboid can be found using the following formula, it is clear that the product of the length, breadth and height is the volume.
Used formula: volume of the cuboid is \[V = lbh\]

Complete step by step answer:
Let us consider, the length, breadth and height of a cuboid is \[l,b,h\]respectively.
So, its volume will be \[V = lbh\]
Again,
The length, breadth and height are increased, decreased and increased by \[1\% ,3\% ,2\% \] respectively.
So, there will be change in the volume.
The length \[l\] is increased by \[1\% .\] so, the new length will be \[\dfrac{{101}}{{100}}l\]
The breadth \[b\] is decreased by \[3\% .\] so, the new breadth will be \[\dfrac{{97}}{{100}}b\]
The height \[h\] is increased by \[2\% .\] so, the new height will be \[\dfrac{{102}}{{100}}h\]
So, the volume will be
\[{V_1} = \dfrac{{101}}{{100}} \times \dfrac{{97}}{{100}} \times \dfrac{{102}}{{100}}lbh\]
To find the change in the volume we will find the difference between these volumes.
So,
\[V - {V_1} = lbh - \dfrac{{101}}{{100}} \times \dfrac{{97}}{{100}} \times \dfrac{{102}}{{100}}lbh\]
We shall solve the above equation to find the increase or decrease in volume, we get,
\[V - {V_1} = \dfrac{{1000000lbh - 999294lbh}}{{1000000}}\]
On subtracting the terms in the equation, we get,
\[V - {V_1} = \dfrac{{704lbh}}{{1000000}}\]
It gives that, \[V > {V_1}\]
Hence, the volume will be decreased when the length, breadth and height will be increased, decreased and increased by \[1\% ,3\% ,2\% \] respectively.

The correct option is (B), The resultant volume of the cuboid Decreases.

Note:
The volume of a three-dimensional shape Cuboid, in general, is equal to the amount of space occupied by the shape cuboid. The term “solid Rectangle” is also known as a cuboid. Because all the faces of a cuboid are rectangular. In a rectangular cuboid, all the angles are at right angles and the opposite faces of a cuboid are equal.
Here we use subtraction to find the increase or decrease in volume another method of substitution can also be used. The method of substitution follows,
We have \[V = lbh\] and\[{V_1} = \dfrac{{101}}{{100}} \times \dfrac{{97}}{{100}} \times \dfrac{{102}}{{100}}lbh\],
From \[{V_1} = \dfrac{{101}}{{100}} \times \dfrac{{97}}{{100}} \times \dfrac{{102}}{{100}}lbh\] we find that \[lbh = \dfrac{{100}}{{101}} \times \dfrac{{100}}{{97}} \times \dfrac{{100}}{{102}}{V_1}\]
Let us substitute l, b, h in V we get,
\[V = \dfrac{{100}}{{101}} \times \dfrac{{100}}{{97}} \times \dfrac{{100}}{{102}}{V_1}\]
That is we can conclude that V is greater than \[{V_1}\], which in turn imply that \[{V_1}\] decreases with the given decrease and increases in length, breadth and height.