Question

The maximum length of a pencil that can be kept in a rectangular box of dimensions \[8cm\times 6cm\times 2cm\] , is ________.
(A). \[2\sqrt{13}\] cm
(B). \[2\sqrt{14}\] cm
(C). \[2\sqrt{26}\] cm
(D). \[10\sqrt{2}\] cm

Answer 1

HINT: - In this question, the most important formula that is to be used to find the length of the body diagonal of a cuboid which is as follows
 \[d=\sqrt{{{l}^{2}}+{{b}^{2}}+{{h}^{2}}}\].
Also, the most important information to solve this problem is the fact that the maximum length inside a cuboid is along its body diagonal only.

Complete step-by-step solution -
As mentioned in the question, we have to find the maximum length of a pencil that can be put inside a cuboid of dimensions that are given in the question and are as follows
 l=8cm, b=6cm, h=2cm

Now, as mentioned in the hint, we can see that the longest length inside a cuboid is the length of its body diagonal, so, we need to calculate its body diagonal.
Therefore, in this question, we need to calculate the body diagonal of the cuboid with the given dimensions.
Now, using the formula for calculating the length of a body diagonal which is given in the hint as well is as follows
\[d=\sqrt{{{l}^{2}}+{{b}^{2}}+{{h}^{2}}}\]
Now, on putting the values in the above formula, we get the following result
\[\begin{align}
  & d=\sqrt{{{8}^{2}}+{{6}^{2}}+{{2}^{2}}} \\
 & d=\sqrt{64+36+4} \\
 & d=\sqrt{104} \\
 & d=2\sqrt{26} cm \\
\end{align}\]
Hence, this is the length of the body diagonal and this is also the length of the maximum sized pencil that can be put inside the given box.


NOTE: - The students can make an error if they don’t know about the formula for calculating the length of the body diagonal as well as if they don’t know that the maximum length inside a cuboid is along its body diagonal only.