In a machine shop, pins of $$15mm$$ diameter are produced at a rate of $$1000$$ per month and the same is consumed at a rate of $$500$$ per month. The production and consumption continue simultaneously till the maximum inventory is reached. Then inventory is allowed to reduced to zero due to consumption. The lot size of production is $$1000.$$ If backlog is not allowed, the maximum inventory levels is

Capacities of production of an item over $$3$$ consecutive months in regular time are $$100,$$ $$100$$ and $$80$$ and in overtime are $$20,$$ $$20$$ and $$40.$$ The demands over those $$3$$ months are $$90,$$ $$130$$ and $$110.$$ The cost of production in regular time and overtime are respectively Rs. $$20$$ per item and Rs. $$24$$ per item. Inventory carrying cost is Rs. $$2$$ per item per month. The levels of starting and final inventory are nil. Backorder is not permitted. For minimum cost of plan, the level of planned production in overtime in the third month is

A thin spherical pressure vessel of $$200mm$$ diameter and $$1mm$$ thickness is subjected to an internal pressure varying from $$4$$ to $$8MPa.$$ Assume that the yield, ultimate and endurance strength of material are $$600,800,400MPa$$respectively. The factor of safety as per Goodman's relation is

A bolted joint in shown below. The maximum shear stress, in $$MPa,$$ in bolts $$A$$ and $$B$$ respectively are