Question

A cube has a total surface area of 384 $c{{m}^{2}}$. Find its volume.

Answer 1

Hint: We need to find the formula for finding the total surface area and the volume of the cube in the form of a variable. We assume the variable and form a quadratic equation involving the given surface area. We solve it to find its value and then from that, we find the value of the volume.

Complete step-by-step solution
Let’s assume the side of a cube is x cm. All the sides of a cube are equal. The cube has 6 surface sides. Each surface area is ${{x}^{2}}$ $c{{m}^{2}}$.
We know that the total surface area of a cube is $6{{x}^{2}}c{{m}^{2}}$.

It’s given that a cube has a total surface area of 384 $c{{m}^{2}}$.
We equate with the formula to find the value of x.
So, $6{{x}^{2}}=384$. We solve it using binary operations.
$\begin{align}
  & 6{{x}^{2}}=384 \\
 & \Rightarrow x=\sqrt{\dfrac{384}{6}}=8 \\
\end{align}$
We are not taking negative sign as a side of a cube can’t be negative.
So, the side of the cube is 8 cm.
Now we have to find the volume of the cube.
The volume of the cube is the measurement of the multiplication of its three dimensions.
So, the volume is ${{x}^{3}}c{{m}^{3}}$. The sides or dimensions are all equal.
We have the value of x. We put the value in the equation to find the volume of the cube.
So, ${{x}^{3}}={{8}^{3}}=512$.
Therefore, the volume of the cube is $512c{{m}^{3}}$.

Note: We need to always remember the general equation of a conic. We have the special form of a conic in the name of the cube and cuboidal but all the theorems and forms are derived from the general forms of conics. Cube is a conic which has each side of equal length.