A manometer is used for the pressure measurement in a closed tank. The three fluids $$f1,f2$$ and $$f3$$ have specific weights $$\gamma ,$$ $$2\gamma ,$$ and $$0.5\gamma ,$$, respectively. The schematic arrangement with manometric readings and other dimensions are shown in the Figure. In order to ensure zero gauge pressure in the tank at the mid-height level $$(h/2),$$ the height of the tank $$h$$ (in $$m$$) is _________________.

A pipeline with variable cross-section contains water with specific weight
$${10^4}\,\,N/{m^3}.$$ The flow conditions at two points $$1$$ and $$2$$ on the axis of the pipe are:
$$\eqalign{ & {P_1} = 3\,\,bar,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{V_1} = 10\,\,m/s \cr & {P_2} = 1\,\,bar,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{V_2} = 20\,\,m/s \cr} $$

Consider frictional losses to be negligible. For no-flow condition between points $$1$$ and $$2$$ (as shown in Figure), if the height $${z_1}$$ from the datum is $$1$$ $$m,$$ then the height $${z_2}$$ (in $$m$$) is _____________ $$\left( {g = 9.81\,\,m/{s^2}} \right)$$

A metallic cylindrical casing of an exhaust pipe has inner radius $$50$$ $$mm$$ and wall thickness $$7$$ $$mm.$$ If the thermal conductivity of the material of the casing is $$50$$ $$W/m$$-$$K,$$ then the thermal resistance of the casing in $$K/kW$$ is ________________ (up to three decimal places).

In a calendar year, the demand forecast of motorbikes for the month of june is $$200.$$ The actual demand of motorbikes for the month of June and July are $$300$$ and $$350,$$ respectively. If single exponential smoothing method with smoothing constant $$0.7$$ is used, then the demand forecast for the month of August is ___________.