In a $$1$$ $$m$$ thick wall, the temperature distribution at a given instant is
$$T\left( x \right) = {c_0} + {c_1}x + {c_2}{x^2}$$ where $$T$$ is in $${}^ \circ C$$ and $$x$$ is in $$m.$$ The constants are: $${c_0} = {800^ \circ }C,$$ $${c_1} = - {250^ \circ }C/m$$ and $${c_2} = - {40^ \circ }C/{m^2}.$$ The thermal conductivity of the wall is $$50$$ $$W/mK$$ and wall area is $$5\,\,{m^2}.$$ If there is a heat source generating uniform volumetric heating at the rate of $$500$$ $$W/{m^3}$$ inside the wall, then the rate of change of energy storage in the wall, in $$kW,$$ is _____________.

A line balancing problem is solved in the context of

A manufacturing line requires $$7.2$$ minutes to make a product. The line has six workstations arranged as per the required sequence of operations. Total production required is $$300$$ products in $$7.5$$ hours. At steady state, the line efficiency, in $$\% $$, is ______________

Arrival of machines for repair in a maintenance shop follows a Poisson distribution at a rate of one per $$18$$ hours. The time to repair follows an exponential distribution with Mean Time To Repair $$(MTTR)$$ of $$14$$ hours. If the productivity loss is Rs.$$22,500$$ per hour, then the total expected loss of productivity due to machine breakdowns, in Rs., is