Question

The length of the shortest face diagonal of a cuboid of dimensions \[5cm\times 4cm\times 3cm\] is:
(a) 4 cm
(b) 5 cm
(c) 6 cm
(d) 7 cm

Answer 1

Hint: In this question, we first need to find the shortest face among all the faces in the cuboid. Then the diagonal of this face will be the hypotenuse which can be calculated by using the formula of diagonal of a rectangle to get the result.

Complete step-by-step answer:
Cuboid: A figure which is surrounded by six rectangular surfaces is called cuboid.
The cuboid shape has six sides called faces. Each face of a cuboid is a rectangle.
Now, let us draw the cuboid with the given dimensions in the question
\[5cm\times 4cm\times 3cm\]

 As already mentioned in the question to find the diagonal of the shortest face
Now, form the given dimensions in the question let us choose the smallest rectangle possible
From,
\[5cm\times 4cm\times 3cm\]
The dimensions of the smallest rectangle possible is
\[4cm\times 3cm\]
Thus, the dimensions of the shortest face will be
\[\Rightarrow 4cm\times 3cm\]
As we already know that in a rectangle of length l and breadth b
The formula of the diagonal of a rectangle is given by
\[diagonal=\sqrt{{{l}^{2}}+{{b}^{2}}}\]
Now, on comparing the values with the above formula we get,
\[l=4,b=3\]
Let us assume the diagonal of the shortest face as d
\[\Rightarrow d=\sqrt{{{l}^{2}}+{{b}^{2}}}\]
Now, on substituting the respective values we get,
\[\Rightarrow d=\sqrt{{{4}^{2}}+{{3}^{2}}}\]
Now, on further simplification we get,
\[\Rightarrow d=\sqrt{25}\]
Now, this can be also written as
\[\Rightarrow d=\sqrt{{{5}^{2}}}\]
\[\therefore d=5cm\]
Hence, the correct option is (b).

Note: It is important to note that the shortest face will be the rectangle which has the least possible dimensions and area because consider any of the rectangle other than that changes the result completely.
Instead of considering it again as a rectangle and assuming some variable to get the diagonal we can directly calculate it from the dimensions we get. It is also important to note that we need to substitute the respective values because neglecting any of the terms changes the result.