Question

The surface area of a cube is $18\dfrac{3}{8}{{m}^{2}}$. Find its volume.

Answer 1

Hint: In this problem we have to find the volume of the cube with the given surface area which is$18\dfrac{3}{8}{{m}^{2}}$. It is in mixed fraction, so we will convert it into normal fraction form. Now we will assume the side of the given cube as $a$ meters. We have the formula for the surface area of the cube which is having side length $a$ as $6{{a}^{2}}$. Now we will equate both the values and simplify the equation, then we will get the side of the cube which is $a$. From the value of side length of the cube we can simply find the volume by using the formula $v={{a}^{3}}$.

Formula use:
1.Surface area of the cube is $6{{a}^{2}}$.
2.Volume of cube is ${{a}^{3}}$.

Complete step by step answer:
Given that, the surface area of the cube is $18\dfrac{3}{8}{{m}^{2}}$.
We can observe that the above value is a mixed fraction, converting it into normal fraction, then we will have
$\begin{align}
  & s=18\dfrac{3}{8}{{m}^{2}} \\
 & \Rightarrow s=\dfrac{18\times 8+3}{8} \\
 & \Rightarrow s=\dfrac{147}{8} \\
\end{align}$
Let us assume the side length of the cube as $a$.
We know that the surface area of the cube which is having side length $a$ is $6{{a}^{2}}$, equating the both the values, then we will get
$6{{a}^{2}}=\dfrac{147}{8}$
Dividing the above equation with $6$ on both sides, then we will have
$\begin{align}
  & \Rightarrow \dfrac{6{{a}^{2}}}{6}=\dfrac{147}{6\times 8} \\
 & \Rightarrow {{a}^{2}}=\dfrac{147}{48} \\
\end{align}$
Cancelling the common factor $3$ in both numerator and denominator, then we will get
\[\Rightarrow {{a}^{2}}=\dfrac{49}{16}\]
Taking square root on both sides of the above equation, then we will get
$\therefore a=\dfrac{7}{4}$
Now the volume of the cube which is having the side length $a$ is ${{a}^{3}}$, hence
$\begin{align}
  & v={{\left( \dfrac{7}{4} \right)}^{3}} \\
 & \Rightarrow v=\left( \dfrac{343}{64} \right){{m}^{3}} \\
\end{align}$

Note:
In this problem we have the three-dimensional shape cube. If they have mentioned the shape as a sphere, then we will use the surface area formula $s=4\pi {{r}^{2}}$. From this we will calculate the value of radius $r$ and use the volume formula $v=\dfrac{4}{3}\pi {{r}^{3}}$.