Question

The volume of a rectangular solid each of whose side, front and bottom faces are 12 sq.cm., 8 sq. cm., and 6 sq. cm. respectively, is:
A. \[576{\text{ }}cu.cm.\]
B. \[24{\text{ }}cu.cm.\]
C. \[9{\text{ }}cu.cm.\]
D. \[104{\text{ }}cu.cm.\]
E. None of these

Answer 1

Hint: We will start by letting length as \[l\], breadth as \[b\] and height as \[h\] of the rectangular solid. We are given the side, front, and bottom as 12, 8, and 6 respectively. As we know that the side of the rectangular solid is defined as \[l \cdot b\] which will be equal to 12 sq.cm and we can define the other two as well in the same manner. Thus, we can multiply all three dimensions to find the volume of the rectangular solid.

Complete step by step answer:

We will first let the length as \[l\], breadth as \[b\], and height as \[h\] of the rectangular solid.
Now, we are given that the side of the rectangular solid which can also be written as \[l \cdot b\] is equal to 12, the front of the rectangular solid which can be written as \[b \cdot h\] is equal to 8 and the bottom of the rectangular side which can be written as \[h \cdot l\] is equal to 6.
Thus, we get,
\[ \Rightarrow l \cdot b = 12,b \cdot h = 8,h \cdot l = 6\]
Now we will multiply all the three obtained values simultaneously,
Thus, we get,
\[
   \Rightarrow lb \cdot bh \cdot hl = 12 \cdot 8 \cdot 6 \\
   \Rightarrow {\left( {lbh} \right)^2} = 12 \cdot 8 \cdot 6 \\
   \Rightarrow {\left( {lbh} \right)^2} = 576 \\
 \]
Next, we will take the square root on both sides of the equation,
We get,
\[
   \Rightarrow lbh = \sqrt {576} \\
   \Rightarrow lbh = 24 \\
 \]
Thus, the volume of the cuboid is calculated by \[ \Rightarrow V = lbh\]. Hence, with this, we can conclude that the volume of the cuboid is \[24c{m^3}\].
Hence, option B is correct.

Note: As the dimensions are not given directly in the form of length, breadth, and height but given in the form of side, front and bottom so, we need to evaluate the volume of the cuboid in this way. As the units are the same, we do not have to change any unit. So, no conversion in units is required. While taking the square root, do it properly. Remember the formula of the volume of a cuboid.