a Question 2 (10 marks) A rechargeable battery for a tool used in production loses its ability to hold a charge as it ages. When considering the time to charge it and manage it, the cost per time unit of use is c(t) = 0.03(t 20)2 + 10 where t is the time since the battery was brand new. A replacement battery can be purchased for $80 and the time to perform the replacement is negligible. a) Determine the optimal age at which to perform a replacement. Include the simplified cost function per unit time, and a graph of the cost function. (4 marks) b) If the client is willing to accept a 5% decrease in optimality for flexibility of replace- ment schedules, what is the recommended range of replacement age? Clearly mark the acceptable range in the graph of the cost function (2 marks) c) Inflationary pressures are pushing the price of batteries up, and we want to ensure that we're within 5% of optimal for battery prices up to $120. What is the recom- mended range of replacement that will satisfy this requirement (5% of optimal) for all battery prices from $80-120? Mark this range in the graph as well. (4 marks) a Question 2 (10 marks) A rechargeable battery for a tool used in production loses its ability to hold a charge as it ages. When considering the time to charge it and manage it, the cost per time unit of use is c(t) = 0.03(t 20)2 + 10 where t is the time since the battery was brand new. A replacement battery can be purchased for $80 and the time to perform the replacement is negligible. a) Determine the optimal age at which to perform a replacement. Include the simplified cost function per unit time, and a graph of the cost function. (4 marks) b) If the client is willing to accept a 5% decrease in optimality for flexibility of replace- ment schedules, what is the recommended range of replacement age? Clearly mark the acceptable range in the graph of the cost function (2 marks) c) Inflationary pressures are pushing the price of batteries up, and we want to ensure that we're within 5% of optimal for battery prices up to $120. What is the recom- mended range of replacement that will satisfy this requirement (5% of optimal) for all battery prices from $80-120? Mark this range in the graph as well. (4 marks)