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

A gear set has a pinion with $$20$$ teeth and a gear with $$40$$ teeth. The pinion runs at $$30rev/s$$ and transmits a power of $$20KW$$. The teeth are on the $${20^ \circ }$$ full-depth system and have a module of $$5mm.$$ The length of the line of action is $$19mm.$$

The center distance for the above gear set in $$mm$$ is