Revenue given by $R(q)=500 q$ and cost is given $C(q)=10,000+5 q^{2}$. At what quantity is profit maximized? What is the profit at this production level?

Step 1 ：First, we need to find the profit function, which is given by the difference between revenue and cost: $P(q) = R(q) - C(q)$. So, we have $P(q) = 500q - (10,000 + 5q^2)$

Step 2 ：Simplify the profit function: $P(q) = -5q^2 + 500q - 10,000$

Step 3 ：To maximize the profit, we can complete the square. First, factor out a $-5$ to get $P(q) = -5(q^2 - 100q)$

Step 4 ：To complete the square, we add $(100/2)^2=2500$ inside the parenthesis and subtract $-5\cdot2500=-12,500$ outside. We are left with the expression $P(q) = -5(q^2 - 100q + 2500) + 12,500 = -5(q-50)^2 + 12,500$

Step 5 ：Note that the $-5(q-50)^2$ term will always be nonpositive since the perfect square is always nonnegative. Thus, the profit is maximized when $-5(q-50)^2$ equals 0, which is when $q=50$. So, the quantity at which profit is maximized is \(\boxed{50}\)

Step 6 ：Now, we need to find the profit at this production level. Substitute $q=50$ into the profit function: $P(50) = -5(50-50)^2 + 12,500 = \boxed{12,500}$