Suppose the firm's production function is given by q equals F left parenthesis K comma L right parenthesis equals K squared L with M P L equals K squared, M P K equals 2 K L.
The price per unit of capital is 20 and the price per unit of labor is 10. Find the cost minimizing quantity of labor to produce 150,000 units of output. Please round to the closest integer.

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a vertical plane with constant angular velocity \omega. write down the lagrangian for a bead of mass m threaded on the rod, using r as your generalized coordinate, where r, \phi are the polar coordinates of the bead. (notice that \phi is not an independent variable since it is fixed by the rotation of the rod tobe \phi = \omega * t) solve lagrange's equation for r(t). what happens if the bead is initially at rest at the origin? if it is released from any point ro > 0, show that r(t) eventually grows exponentially. explain your results in terms of the centrifugal force m* \omega ^2 *r.