If a short$-$run Cobb$-$Douglas production function is $q=L^{0\mathrm{.}75}{K}^{0\mathrm{.}25}$ and its factor prices of labor, i$.$e$.,w=10$ and capital, i$.$e$.,r=20,$ and its capital $(K)$ is fixed at $50,$ then what are its short$-$run average fixed cost, average variable cost, and its marginal cost functions? Remember, all these will be a function of L$.(xx5=15$ points$)$

A beef processing plant uses two inputs to produce its products: workers $($ L $)$ and canning machines $($M$).$ The rent of the machine per day is $$5600$ per day, and the worker's wage is $$200$ per day. The marginal product of the machine is $400,$ while the marginal product of labor is $100.$ Now answer the following questions:

$($i$)$ Is the beef processing plant minimizing cost at this point? Explain. $(5+5=10$ points$)$

$($ii$)$ How can the beef processing plant minimize cost? $(5$ points$)$

A retailer uses labor $($ L $)$ and scanning machines $(M)$ for its business. The production function of the retailer is $q=L^{0\mathrm{.}65}{M}^{0\mathrm{.}35}.$ The factor costs are the following: wage $(w)=$$$15,$ rent $(r)=$$$500.$

$($i$)$ Calculate the minimum$-$cost combination of inputs for this retailer if the output produced is $100$ units. $(10$ points$)$

$($ii$)$ What is the actual cost paid by this retailer if he uses the minimum$-$cost combination of inputs? $(10$ points$)$