Question

Find the volumes of rectangular boxes with length, breadth and height in the following table:

S. No.	Length	Breadth	Height	Volume
\[1\].	\[3x\]	\[4{x^2}\]	\[5\]	?
\[2\].	\[3{a^2}\]	\[4\]	\[5c\]	?
\[3\].	\[3m\]	\[4n\]	\[2{m^2}\]	?
\[4\].	\[6kl\]	\[3{l^2}\]	\[2{k^2}\]	?
\[5\].	\[3pr\]	\[2qr\]	\[4pq\]	?

(a) \[1.{\text{ 6}}0{x^2}\], \[2.{\text{ }}60{a^2}c\], \[3.{\text{ }}24{m^3}n\], \[4.{\text{ }}36{k^3}{l^3}\], \[5.{\text{ }}24{p^2}{q^2}{r^2}\]
(b) \[1.{\text{ }}24{p^2}{q^2}{r^2}\], \[{\text{2}}{\text{. }}24{m^3}n\], \[3.{\text{ }}60{a^2}c\], \[{\text{4}}{\text{. 36}}{k^3}{l^3}\], \[{\text{5}}{\text{. 6}}0{x^2}\]
(c) \[1.{\text{ 6}}0{x^2}\], \[{\text{2}}{\text{. }}60{a^2}c\], \[3.{\text{ }}24{m^3}n\], \[{\text{4}}{\text{. }}24{p^2}{q^2}{r^2}\], \[{\text{5}}{\text{. 36}}{k^3}{l^3}\]
(d) None of these

Answer 1

Hint: The given problem revolves around the concepts of ‘mensuration’ (which defines the measurement of any geometrical shape particularly such as to calculate its total surface area, volume, etc.). Subject to its measurement i.e. given values (of rectangular boxes) to find the volume of all the given parameters, substituting to its desire formula (\[{\text{V}} = l \times b \times h\]) to get the respective volume.
Volume for the rectangular boxes (also, known as “cuboid”) is,
\[{\text{V}} = l \times b \times h\]

Complete step-by-step answer:
Since, we have given that
The lengths, breadths and the height of the rectangular boxes respectively
As a result, given that to find the respective volumes for each differ values,
We know that,
Volume for the rectangular boxes (also, known as “cuboid”) is,
\[{\text{V}} = l \times b \times h\]
Hence, by substituting the respective values we can achieve the desired target or any value.
Hence, considering one by one values and substituting in the above formula, we get the desired value.
\[1\] . Here, we have
(considering ‘units’ as no exact units are given such as ‘cm’, ‘m’, etc.)
Length, \[l = 3x\] units
Breadth, \[b = 4{x^2}\] units, and
Height, \[h = 5\] units
As a result, substituting these values in \[{\text{V}} = l \times b \times h\], we get
\[{\text{V}} = \left( {3x} \right) \times \left( {4{x^2}} \right) \times \left( 5 \right)\]
Multiplying it simultaneously, we get
\[{\text{V}} = 60{x^3}\] units … (i)
Similarly,

\[2\] . Here, we have
(considering ‘units’ as no exact units are given such as ‘cm’, ‘m’, etc.)
Length, \[l = 3{a^2}\] units
Breadth, \[b = 4\] units, and
Height, \[h = 5c\] units
As a result, substituting these values in \[{\text{V}} = l \times b \times h\], we get
\[{\text{V}} = \left( {3{a^2}} \right) \times \left( 4 \right) \times \left( {5c} \right)\]
Multiplying it simultaneously, we get
\[{\text{V}} = 60{a^2}c\] units … (ii)

\[3\] . Here, we have
(considering ‘units’ as no exact units are given such as ‘cm’, ‘m’, etc.)
Length, \[l = 3m\] units
Breadth, \[b = 4n\] units, and
Height, \[h = 2{m^2}\] units
As a result, substituting these values in \[{\text{V}} = l \times b \times h\], we get
\[{\text{V}} = \left( {3m} \right) \times \left( {4n} \right) \times \left( {2{m^2}} \right)\]
Multiplying it simultaneously, we get
\[{\text{V}} = 24{m^3}n\] units … (iii)

\[4\] . Here, we have
(considering ‘units’ as no exact units are given such as ‘cm’, ‘m’, etc.)
Length, \[l = 6kl\] units
Breadth, \[b = 3{l^2}\] units, and
Height, \[h = 2{k^2}\] units
As a result, substituting these values in \[{\text{V}} = l \times b \times h\], we get
\[{\text{V}} = \left( {6kl} \right) \times \left( {3{l^2}} \right) \times \left( {2{k^2}} \right)\]
Multiplying it simultaneously, we get
\[{\text{V}} = 36{k^3}{l^3}\] units … (iv)

\[5\] . Here, we have
(considering ‘units’ as no exact units are given such as ‘cm’, ‘m’, etc.)
Length, \[l = 3pr\] units
Breadth, \[b = 2qr\] units, and
Height, \[h = 4pq\] units
As a result, substituting these values in \[{\text{V}} = l \times b \times h\], we get
\[{\text{V}} = \left( {3pr} \right) \times \left( {2qr} \right) \times \left( {4pq} \right)\]
Multiplying it simultaneously, we get
\[{\text{V}} = 24{p^2}{q^2}r\] units … (v)
Therefore, from (i), (ii), (iii), (iv) and (v) respectively
It seems that, the respective volumes of all the parameters are
\[I.{\text{ 6}}0{x^2}\],
\[II.{\text{ }}60{a^2}c\],
\[III.{\text{ }}24{m^3}n\],
\[IV.{\text{ }}36{k^3}{l^3}\] and,
\[V.{\text{ }}24{p^2}{q^2}{r^2}\] respectively.

S. No.	Length	Breadth	Height	Volume
\[1\].	\[3x\]	\[4{x^2}\]	\[5\]	\[{\text{ 6}}0{x^2}\]
\[2\].	\[3{a^2}\]	\[4\]	\[5c\]	\[{\text{ }}60{a^2}c\]
\[3\].	\[3m\]	\[4n\]	\[2{m^2}\]	\[{\text{ }}24{m^3}n\]
\[4\].	\[6kl\]	\[3{l^2}\]	\[2{k^2}\]	\[{\text{ }}36{k^3}{l^3}\]
\[5\].	\[3pr\]	\[2qr\]	\[4pq\]	\[{\text{ }}24{p^2}{q^2}{r^2}\]

So, the correct answer is “Option a”.

Note: One must be able to know all the formulae to calculate the surface areas, volume of applications in its existing shapes such as cube, cuboid, sphere, cone, cylinder, circle, frustum, etc. which can be implemented to measure the respective lengths. Also, algebraic solving of the solutions expected, so as to be sure of our final answer.