A long slender metallic rod of length $$L$$ is used as a fin. As shown in the figure below, the left end of the fin is kept at a constant temperature $${t_b}.$$ The fin loses heat by convection to the atmosphere which is at a temperature ta $${t_a}\,\,\left( {{t_a} < {t_b}} \right).$$ Four options of temperature profiles are shown. Identify the correct option.

Consider a network with nodes $$1, 2, 3, 4, 5$$ and $$6.$$ The nodes are connected with directed arcs as shown in the table below. The respective costs (in $$INR$$) incurred while traversing the directed arcs are also mentioned.

The second shortest path from node $$1$$ to node $$6$$ (i.e. the path that has the second least total cost and does not use any part of the shortest path) has a total cost (in $$INR$$) of

The throughput rate of a production system is $$20$$ units per hour. The average flow time is $$30$$ minutes and the cycle time is $$3$$ minutes. The average inventory (in units) in the system is

Two flat steel sheets, each of $$2.5$$ $$mm$$ thickness, are being resistance spot welded using a current of $$6000$$ $$A$$ and weld time of $$0.2$$ $$s.$$ The contact resistance at the interface between the two sheets is $$200\,\,\mu \Omega $$ and the specific energy to melt steel is $$10 \times {10^9}\,\,J/{m^3}.$$ A spherical melt pool of diameter $$4$$ $$mm$$ is formed at the interface due to the current flow. Consider that electrical energy is completely converted to thermal energy. The ratio of the heat used for melting to the total resistive heat generated is _____________