For flow through a pipe of radius $$R,$$ the velocity and temperature distribution are as follows:
$$U\left( {r,x} \right) = {C_1}$$ and $$T\left( {r,x} \right) = {C_2}\left[ {1 - {{\left( {{r \over R}} \right)}^3}} \right],$$
where $${C_1}$$ and $${C_2}$$ are constants. The bulk mean temperature is given by

$${T_m} = {2 \over {{U_m}{R^2}}}\int\limits_0^R {u\left( {r,x} \right)T\left( {r,x} \right)rdr,} $$
with $${{U_m}}$$ being the mean velocity of flow. The value of $${T_m}$$ is

Following data refers to the activities of a project, where, node $$1$$ refers to the start and node $$5$$ refers to the end of the project.

The critical path $$(CP)$$ in the network is

For a canteen, the actual demand for disposable cups was $$500$$ units in January and $$600$$ units in February. The forecast for the month of January was $$400$$ units. The forecast for the month of March considering smoothing coefficient as $$0.75$$ is_______________

A machine element is subjected to the following bi-axial state of stress: $$\,\,{\sigma _x} = 80MPa;\,\,{\sigma _y} = 20MPa;\,\,{\tau _{xy}} = 40MPa.$$ If the shear strength of the material is $$100 MPa,$$ the factor of safety as per Tresca’s maximum shear stress theory is