Question

The surface area of a cuboid is 164 cm\[^2\] and the length, the height of the cuboid is 10 cm and 4 cm respectively. What will be the volume of the cuboid?
A. 120 cm\[^3\]
B. 100 cm\[^3\]
C. 110 cm\[^3\]
D. None of these

Answer 1

Hint: First, we will assume that breadth of the cuboid is \[b\] cm. Then we will calculate the value of b by using the formula of the surface area of cuboid which is \[2\left( {lb + bh + hl} \right)\], where \[l\] is the length, \[b\] is the breath and \[h\] is the height of the cuboid and then we will use the formula to calculate the volume of the cuboid is \[lbh\], where \[l\] is the length, \[b\] is the breadth and \[h\] is the height of cuboid to find the required value.

Complete step by step answer:

We are given that the surface area of a cuboid is 164 cm\[^2\] and the length \[l\], height \[h\] of the cuboid is 10 cm and 4 cm respectively.

Let us assume that the breadth of the cuboid is \[b\] cm.

We know that the formula to calculate the surface area of cuboid is \[2\left( {lb + bh + hl} \right)\], where \[l\] is the length, \[b\] is the breath and \[h\] is the height of the cuboid.

From the above formula of the surface area of the cuboid and the value of surface area, we get

\[ \Rightarrow 2\left( {lb + bh + hl} \right) = 164\]

Substituting the value of \[l\] and \[h\] in the above equation, we get
\[
   \Rightarrow 2\left( {10b + 4b + 40} \right) = 164 \\
   \Rightarrow 2\left( {14b + 40} \right) = 164 \\
   \Rightarrow 28b + 80 = 164 \\
 \]

Subtracting the above equation by 80 on each of the sides, we get
\[
   \Rightarrow 28b + 80 - 80 = 164 - 80 \\
   \Rightarrow 28b = 84 \\
 \]

Dividing the above equation by 28 on both sides, we get
\[
   \Rightarrow \dfrac{{28b}}{{28}} = \dfrac{{84}}{{28}} \\
   \Rightarrow b = 3 \\
 \]

Thus, the breadth of the cuboid is 3 cm.

We will use the formula to calculate the volume of the cuboid is \[lbh\], where \[l\] is the length, \[b\] is the breadth and \[h\] is the height of the cuboid.

Substituting the values of \[l\], \[b\] and \[h\] in the above formula, we get
\[
   \Rightarrow {\text{Volume}} = 10 \times 3 \times 4 \\
   \Rightarrow {\text{Volume}} = 120{\text{ c}}{{\text{m}}^3} \\
 \]
Thus, the volume of the cuboid is 120 cm\[^3\].
Hence, option A is correct.

Note: In solving these types of questions, you should be familiar with the formula of the surface area of cuboid and volume of a cuboid. Some students use the formula of the curved surface area instead of the total surface area of the cylinder, which is wrong. We need to use the given conditions and values given in the question properly, and substitute the values in this formula, to find the required value.