Question

A rectangular solid has three faces with areas of 28, 20, and 35 square centimeters. Calculate the volume of this solid
A.83 cubic cm
B.140 cubic cm
C.166 cubic cm
D.196 cubic cm

Answer 1

Hint: Here, we will use the area of the faces of a rectangular solid to find the volume of the rectangular solid. First, we take the area of the first face as the product of length and width. Then, we take the area of the first face as the product of height and width and the third face as the product of length and height. We will multiply all these areas and solve them further to get the required volume.

Formula Used:
We will use the following formula:
1.Area of the first face of a rectangular solid is given by \[{A_1} = l \times w\]
2.Area of the second face of a rectangular solid is given by \[{A_2} = w \times h\]
3.Area of the third face of a rectangular solid is given by \[{A_3} = h \times l\]
4.Volume of a rectangular solid is given by \[V = l \times w \times h\]

Complete step-by-step answer:
We are given with the rectangular solid has three faces with areas of 28, 20, and 35 square centimeters.
Area of the first face of a rectangular solid is given by \[{A_1} = l \times w\]
Area of the first face \[ = 28c{m^2}\]
\[ \Rightarrow l \times w = 28c{m^2}\] ……………………………………………………………………\[\left( 1 \right)\]
Area of the second face of a rectangular solid is given by \[{A_2} = w \times h\]
Area of the second face \[ = 20c{m^2}\]
\[ \Rightarrow \] \[w \times h = 20c{m^2}\] ……………………………………………………………………..\[\left( 2 \right)\]
Area of the third face of a rectangular solid is given by \[{A_3} = h \times l\]
Area of the third face \[ = 35c{m^2}\]
\[ \Rightarrow \] \[h \times l = 35c{m^2}\] ………………………………………………………………………..\[\left( 3 \right)\]
Volume of a rectangular solid is given by \[V = l \times w \times h\]
Multiplying equation \[\left( 1 \right)\], equation \[\left( 2 \right)\] and equation\[\left( 3 \right)\], we get
\[l \times w \times w \times h \times h \times l = 28 \times 20 \times 35\]
\[ \Rightarrow {l^2} \times {w^2} \times {h^2} = 28 \times 20 \times 35\]
By simplification, we get
 \[ \Rightarrow {\left( {l \times w \times h} \right)^2} = 4 \times 7 \times 5 \times 4 \times 7 \times 5\]
By taking square root on both the sides, we get
\[ \Rightarrow \left( {l \times w \times h} \right) = \sqrt {4 \times 7 \times 5 \times 4 \times 7 \times 5} \]
\[ \Rightarrow \left( {l \times w \times h} \right) = 4 \times 7 \times 5\]
Multiplying the terms, we get
\[ \Rightarrow \left( {l \times w \times h} \right) = 140c{m^3}\]
Therefore, the volume of a rectangular solid is 140 cubic centimeters.
Thus, option (B) is the correct answer.

Note: We should know that the area of the face of the rectangular solid is similar to the area of the rectangle since the rectangular solid has six rectangles on all its sides and we are given an area of three faces out of six faces. Volume of a rectangular solid can also be obtained by the area of the rectangular faces and is defined as the quantity of substance that an enclosed container can hold. A rectangular solid is defined as a three dimensional object with six sides of which are rectangles which is similar to Cuboid.