For the given cost function $C(x)=52900+200 x+x^{2}$, First, find the average cost function. Use it to find:
a) The production level that will minimize the average cost $x=$
b) The minimal average cost
$\$$

Step 1 ：The average cost function is given by $C(x)/x$. So, we have $AC(x) = \frac{52900+200x+x^{2}}{x}$

Step 2 ：Simplify the average cost function to get $AC(x) = \frac{52900}{x}+200+x$

Step 3 ：To find the production level that will minimize the average cost, we need to find the derivative of the average cost function and set it equal to zero. So, we have $AC'(x) = -\frac{52900}{x^{2}}+1$

Step 4 ：Setting the derivative equal to zero gives $-\frac{52900}{x^{2}}+1=0$

Step 5 ：Solving for $x$ gives $x = \sqrt{52900}$

Step 6 ：So, the production level that will minimize the average cost is $x = \boxed{\sqrt{52900}}$

Step 7 ：To find the minimal average cost, we substitute $x = \sqrt{52900}$ into the average cost function. So, we have $AC(\sqrt{52900}) = \frac{52900}{\sqrt{52900}}+200+\sqrt{52900}$

Step 8 ：Simplify to get $AC(\sqrt{52900}) = 2\sqrt{52900}$

Step 9 ：So, the minimal average cost is $\boxed{2\sqrt{52900}}$