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

The correct option is 9/2.

Step 1: Understand the definition of a fourth proportional

If four numbers x, y, z, and w are in proportion, then w is called the fourth proportional to x, y, and z. The proportion relationship is expressed as:

$\frac{x}{y}=\frac{z}{w}$

Step 2: Set up the equation using the given terms

In this problem, the first three terms are 4, $a$, and 16$a$, and the fourth proportional is 81. Substituting these into the proportion formula gives:

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

Step 3: Solve for a

Cross-multiply the terms of the equation:

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

$324=16a^2$

Divide both sides by 16 to isolate $a^2$:

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

Simplify the fraction by dividing the numerator and denominator by 4:

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

Take the positive square root of both sides:

$a=\sqrt{\frac{81}{4}}=\frac{9}{2}$

Thus, the value of a is 9/2.