The daily cost of producing $q$ in a manufacture company is $$c = 5000 + 100q.$$ 1) If each item is sold for $\$200$, how can I calculate the minimum quantity without any loss?
$$200q - (5000 + 100q) = 100q +5000 = 50.$$ is this correct?
2) Also, if the price is up by $10$ Dollars per unit, how can I calculate the break even point? $$100 + 10 = 110, 110q - (5000 + 100q) = 500$$ is this correct?
You break even when the sales income for the quantity $q$ equals the cost of production for the same quantity $q$.
Case 1
$$cost=5000+100q$$ $$SalesIncome=200q$$
Break point is for $q$ to satisfy:
%%
$$Cost=SalesIncome$$
$$5000+100q = 200q$$
so, $$5000=100q$$
and hence $$q=5000/100=50$$
That is you need to sell 50 items to break even.
Case 2
When the price goes up by $10$ Dollars, the price becomes $210$
Break point is for $q$ to satisfy: $$5000+100q = 210q$$
so, $$5000+100q = 210q-100q=10q$$
Hence the quantity is $500$ units to break even.
This picture shows case 1: