Altitude

The surface area of a regular tetrahedron is

$A=s^2\sqrt{3}$

$100\sqrt{3}=s^2\sqrt{3}$ $\implies$ $s=10$ $\implies$ $\dfrac{s}{2}=5$

$cos~30=\frac{5}{x}$ but $cos~30=\dfrac{\sqrt{3}}{2}$

So,

$\frac{\sqrt{3}}{2}=\frac{5}{x}$ $\implies$ $x=\dfrac{10}{\sqrt{3}}$

By Pythagorean Theorem,

$h^2=s^2-x^2$ $\implies$ $h^2=100-\dfrac{100}{3}$ $\implies$ $h=\dfrac{5(2)\sqrt{2}}{\sqrt{3}}$

$\large\therefore$ $A+B+C=5+2+3=10$