$\backslash (6-34\backslash )$ Sell or Process Further

ConAgra produces meat products with brand names such as Healthy Choice, Armour, and Butterball. Suppose one of the company's plants processes beef cattle into various products. For simplicity, assume that there are only three products: steak, hamburger, and hides, and that the average steer costs $\backslash (\backslash$$ $700\backslash ).$ The three products emerge from a process that costs $\backslash (\backslash$$ $100\backslash )$ per steer to run, and output from one steer can be sold for the following net amounts:

Assume that each of these three products can be sold immediately or processed further in another ConAgra plant. The steak can be the main course in frozen dinners sold under the Healthy Choice label. The vegetables and desserts in the $400$ dinners produced from the $100$ pounds of steak would cost $\backslash (\backslash$$ $110\backslash ),$ and production, sales, and other costs for the $400$ meals would total $\backslash (\backslash$$ $330\backslash ).$ Each meal would be sold wholesale for $\backslash (\backslash$$ $2\mathrm{.}10\backslash ).$

The hamburger could be made into frozen Salisbury steak patties sold under the Armour label. The only additional cost would be a $\backslash (\backslash$$ $200\backslash )$ processing cost for the $500$ pounds of hamburger. Frozen Salisbury steaks sell wholesale for $\backslash (\backslash$$ $1\mathrm{.}70\backslash )$ per pound.

The hide can be sold before or after tanning$.$ The cost of tanning one hide is $\backslash (\backslash$$ $80\backslash ),$ and a tanned hide can be sold for $\backslash (\backslash$$ $170\backslash ).$

$1.$ Compute the total profit if all three products are sold at the split$-$off point.

$2.$ Compute the total profit if all three products are processed further before being sold.

$3.$ Which products should be sold at the split$-$off point? Which should be processed further?

$4.$ Compute the total profit if your plan in number $3$ is followed.