A manufacturer of batteries offers a two-year limited warranty. If the battery fails for any reason during the first two years, it is replaced. The time to failure, T, has the density function f(t)=te^(-(t^2)/a) for t >0.
(a) If a = 2
1) What percentage of the batteries is expected to fail within the warranty period?
2) The manufacturing cost of a battery is $75, and the profit per sale is $25 at the time of sale, i.e, the selling price is $100. What is the effect of warranty replacements on profit if during the warranty period a failed battery is replaced at no cost to the customer?
(b) What value of is required for a net profit of $10 per sale?