If third proportional of 8 and 48 be x, then what is the value of x?

The correct option is 288.

To find the third proportional of two numbers, we can use the definition of continued proportion. If three numbers $a$, $b$, and $c$ are in continued proportion, then we can write the relationship as:

$\frac{a}{b}=\frac{b}{c}$

Here, $c$ is called the third proportional of $a$ and $b$.
By cross-multiplying, we get:
$a·c=b^2$
Which gives the formula for the third proportional $c$:
$c=\frac{b^2}{a}$

In this problem, the given numbers are $a=8$ and $b=48$, and the third proportional is represented by $x$.
Substituting these values into the formula, we have:

$x=\frac{{48}^2}{8}$

Now, let's calculate the value step-by-step:
${48}^2=48·48$
So,
$x=\frac{48·48}{8}$

We can simplify this by dividing 48 by 8 first:
$\frac{48}{8}=6$

Now, multiply the result by the remaining 48:
$x=6·48$
$x=288$

Thus, the value of the third proportional $x$ is 288.