Write an equation describing the relationship of the given variables.

$y$ varies directly as the fourth power of $x$ and when $x=3,y=162$.

Forming the required equation.

Step 1: Relation between $x$ and $y$.

Given, $y$ varies directly as the fourth power of $x$.

We can write that as,

$y\propto x^4$

Converting variation to an equation,

$y=k{x}^4$

Step 2: Finding $\mathrm{k}$

Now, substituting $x=3,y=162$ in the above equation,

$$\begin{gathered} \Rightarrow 162=\mathrm{k}{\left(3\right)}^4 \\ \Rightarrow 162=\mathrm{k}\left(81\right) \\ \Rightarrow \mathrm{k}=\frac{162}{81} \\ \Rightarrow \mathrm{k}=2 \end{gathered}$$

Substituting the value of $\mathrm{k}$ in $\mathrm{y}={\mathrm{kx}}^4$.

$\Rightarrow \mathrm{y}=2{\mathrm{x}}^4$

Therefore, the equation describing the given relationship is $y=2x^4$.