The above image shows a dam and an outlet pipe. All dimensions are given in figure itself. The bold segment indicates an elliptical plate acting as valve. It is inclined with the horizontal plane at an angle \(\alpha \).

The torque necessary to hold the valve in place can be expressed as \( \tau =\kappa \rho g{ h }^{ \eta }\cfrac { { R }^{ \mu } }{ \sin ^{ \lambda }{ \alpha } } \) where \( \kappa , \lambda , \mu , \eta \) are non negative numerical constants.

Find value of \( \kappa + \lambda + \mu + \eta \).

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