Sappose that a firm's production function is $q=10L^{\frac{1}{2}}{k}^{\frac{1}{2}}.$ The cost per unit of labour is $20$ and the cost of a unit of capital is $80.$ If the marginal rate of technical substitution is $\frac{K}{1}.$

$($i$)$ Find the optimal level of capital and lahour required to produce $140$ units of output.

$7$i$)$ Determine the cost function.

ender what condition would the long run total cost curve be positively sloped straight line throught the origin?

What would be the shapes of long run average cost and long run marginal cost curves?

Would this be consistent with the u$-$shaped short run average cost curv