An elevator rail is assumed to meet specifications if its diameter is between $0\mathrm{.}98$ and $1\mathrm{.}01$ inches. Each year a

company produces $100,000$ elevator rails. For a cost of $$\frac{10}{{}^2}$ per year the company can rent a machine that

produces elevator rails whose diameters have a standard deviation of $.($The idea is that the company must pay

more for a smaller variance.$)$ Each such machine will produce rails having a mean diameter of $1$ inch. Any rail that

does not meet specifications must be reworked at a cost of $$12.$ Assume that the diameter of an elevator rail follows

a normal distribution.

Round your answers to three decimal places, if necessary.

a$.$ What standard deviation $($within $0\mathrm{.}001$ inch$)$ minimizes the annual cost of producing elevator rails? You do not

need to try standard deviations in excess of $0\mathrm{.}02$ inch.

b$.$ For your answer in part a$,$ one elevator rail in $1000$ will be at least how many inches in diameter?