Question 31363
Well, first, let's see the definition of an apothem of a regular polygon, of which, an equilateral triangle is certainly an example:
"An apothem of a regular polygon is a line drawn from its centre perpendicular to one of its sides" This is also the radius of the inscribed circle.

The formula for finding  an apothem of a regular polygon is:
{{{r = (1/2)(s)cot(180/n)}}}
Where:
r = is the length of the apothem.
s = the length of one side of the regular polygon (equilateral triangle).
n = the number of sides in the regular polygon (3).

One minor problem is...you don't know the length of one side (s) of the equilateral tringle!

Not to worry however because you do know the area and you can use Heron's formula for finding the length of the side of the equilateral triangle.

Heron's formula, which gives the area of a triangle as a function of the length of the sides is:

{{{A = sqrt(s(s-a)(s-b)(s-c))}}}
Where:
s = the semi-perimeter of the triangle.
a, b, c, are the lengths of the sides of the triangle.

But, in an equilateral triangle, a = b = c and {{{s = (a+b+c)/2}}} = {{{3a/2}}}

So, let's find the length (a) of one side of the triangle using Heron's formula {{{A = sqrt(s(s-a)(s-b)(s-c))}}}and the known area of the triangle{{{300sqrt(3)}}}
Rewrite Heron's formula for the case of an equilateral triangle where {{{s = 3a/2}}}
{{{A = sqrt((3a/2)((3a/2)-a)^3)}}} Simplify.
{{{A = sqrt((3a/2)(a/2)^3)}}}

{{{A = sqrt((3a/2)(a^3/8))}}}
{{{A = sqrt(3a^4/16)}}}
{{{A = (a^2/4)sqrt(3)}}} But the area of the triangle is given as {{{A = 300sqrt(3)}}}, so:
{{{300sqrt(3) = (a^2/4)sqrt(3)}}} Simplifying, we get:
{{{300 = a^2/4}}} Solving for a, the length of the side of the triangle:
{{{a^2 = 4(300)}}}
{{{a^2 = 1200}}} Taking the square root of both sides.
{{{a = 20sqrt(3)}}}

Now we can substitute this for s in the formula for the apothem.

{{{r = (1/2)20sqrt(3)cot(180/3)}}} Simplifying.
{{{r = 10sqrt(3)cot(60)}}}
{{{r = 10sqrt(3)(0.577)}}}
{{{r = 5.77sqrt(3)}}} This is the length of the apothem.