In a company we have $20$ similar machines . Number of damaged machines has binomial distribution with $p = 0.3$. In other words, $$\mathbb{P}[X = k] = { 20 \choose k}0.3^k 0.7^{20-k}.$$ In order to repairing damaged machines , we should pay $150$ for each of them . The company is able to buy new machines and replace with the damaged machines but the cost for maintaining of reserved machines is $200$ for each . Find number of reserved machines that company should buy so that the cost be minimum .
My try : Let $k$ be number of damaged machines , $k'$ reserved machines and $C$ cost . This algorithm finds the cost :
if $k \gt k'$ $$C = 50k + 150(k - k')$$ if $k = k'$ $$C = 50k$$ if $k \lt k'$ $$C = 50k + 200(k' - k)$$
And I got stuck here , thanks in advance !