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.

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

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

In linear gas tungsten arc welding of two plates of the same material, the peak temperature $$T$$ (in $$K$$) is given as ܶ$$T = {{{c_1}q} \over \alpha },$$ where $$q$$ is the heat input per unit length (in $$J/m$$) of weld, $$\alpha $$ is the thermal diffusivity (in $${m^2}/s$$) of the plate materials and $${C_1}$$ is a constant independent of process parameters and material types. Two welding cases are given below.

where, $$V$$ is welding voltage, $$I$$ is welding current, $$v$$ is welding speed, and $$k,$$ $$\rho $$ and $$C$$ refer to the thermal conductivity, the density and the specific heat of the plate materials, respectively. Consider that electrical energy is completely converted to thermal energy. All other conditions remain same.

The ratio of the peak temperature in Case $${\rm I}$$ to that in Case $${\rm I}{\rm I}$$ is