The fourth proportional of $4$, $a$, and $16a$ is $81$. What is the value of $a$?

If four quantities $p,q,r$, and $s$ are in proportion, then their relationship is defined as:
$$\frac{p}{q}=\frac{r}{s}$$

Here, the given terms are $4$, $a$, $16a$, and their fourth proportional is $81$. Substituting these values into the proportionality formula:

$$\frac{4}{a}=\frac{16a}{81}$$

Cross-multiplying the terms:

$$4\times 81=a\times 16a$$

$$324=16a^2$$

$$a^2=\frac{324}{16}$$

Simplifying the fraction by dividing both the numerator and the denominator by $4$:

$$a^2=\frac{81}{4}$$

Taking the positive square root on both sides:

$$a=\frac{9}{2}$$