Problem $6$:

You are hired as consultant for a company that manufactures two types of wood$-$burning stoves: a freestanding model and a fireplace$-$insert model. The cost for producing $x$ freestanding and $y$ fireplace$-$insert stoves is

$C=32\sqrt[\phantom{2}]{xy}175x205y1050$

$($a$)$ Find, for your employer, the marginal costs $(C_x,C_y)$ when $x=80$ and $y=20.$

$($b$)$ Your employer wants to know, when additional production is required, which model stove results in the cost increasing at a faster rate. He$/$she also wants you to show how this can be determined from the cost model.