Question

The surface area of the three coterminous faces of a cuboid are $6c{{m}^{2}},15c{{m}^{2}},10c{{m}^{2}}$ respectively. Find the volume of the cuboid.
\[\begin{align}
  & A.30c{{m}^{3}} \\
 & B.20c{{m}^{3}} \\
 & C.40c{{m}^{3}} \\
 & D.35c{{m}^{3}} \\
\end{align}\]

Answer 1

Hint: In this question, we are given a surface area of three faces of a cuboid and we have to find volume of the cuboid. Since, surface area and volume are obtained from three terms: length, breadth and height, we will use the given surface area to find length, breadth and height in the form of formula to find volume of cuboid. Volume of cuboid is given by $V=l\times b\times h$ where, l is length, b is breadth and h is height of cuboid.

Complete step-by-step answer:
Here, we are given surface area of three coterminous faces of cuboid as $6c{{m}^{2}},15c{{m}^{2}},10c{{m}^{2}}$ respectively. Let us draw a diagram of cuboid to understand the question easily.

Here, we will take l as length, b as breadth and h as height.
As we can see, the area of the face of the cuboid containing length and breadth is $15c{{m}^{2}}$. So, the surface area of the rectangle (face) is given by $\text{length}\times \text{breadth}$. Therefore, $l\times b=15c{{m}^{2}}$.
Similarly, the area of the face of the cuboid containing breadth and height is $6c{{m}^{2}}$. Therefore, $b\times h=6c{{m}^{2}}$.
And the area of the face of the cuboid containing length and height is $10c{{m}^{2}}$. Therefore, $l\times h=10c{{m}^{2}}$.
Hence, we get $lb=15c{{m}^{2}},bh=6c{{m}^{2}},lh=10c{{m}^{2}}$.
Now, we need to find the volume of the cuboid. We know that, volume of the cuboid is given by $V=l\times b\times h=lbh$.
Hence, we need to find the value of lbh. For this, let us multiply lb, bh, lh as found earlier we get:
\[\begin{align}
  & l\times b\times b\times h\times l\times h=15\times 6\times 10 \\
 & \Rightarrow {{l}^{2}}\times {{b}^{2}}\times {{h}^{2}}=15\times 6\times 10 \\
\end{align}\]
Taking square common from left side we get:
\[{{\left( lbh \right)}^{2}}=900\]
Taking square root both sides we get:
\[lbh=\sqrt{900}\]
Since 900 can be written as $3\times 3\times 10\times 10$. Therefore,
\[\sqrt{900}=\sqrt{3\times 3\times 10\times 10}=3\times 10=30\]
Hence, $lbh=30c{{m}^{3}}$.
Since lbh represents the volume of cuboid. Hence, the volume of cuboid is equal to $30c{{m}^{3}}$.

So, the correct answer is “Option A”.

Note: Students should never forget to put units after finding volume. If they can't remember units they should use units of given measurements to find required unit as below:
\[\begin{align}
  & {{\left( lbh \right)}^{2}}=15c{{m}^{2}}\times 6c{{m}^{2}}\times 10c{{m}^{2}} \\
 & \Rightarrow {{\left( lbh \right)}^{2}}=900c{{m}^{6}} \\
 & \Rightarrow lbh=\sqrt{900c{{m}^{6}}} \\
 & \Rightarrow lbh=30c{{m}^{3}} \\
\end{align}\]
This way units can be written without getting confused. Cubic units are used for volume and square units are used for areas.