Show all your work including assumptions, statements, arguments, etc.$|$

$(30$ points$)$ XYZ is a toy manufacturer. It manufactures both toy components and finished

toys. For a specific type of toys, the manufacturer can sell the components to an assembly

company for $$10$ per set, or assemble the toys itself. The finished toys can be sold for $$16$

each. XYZ has a total capacity of $500,000$ hours, which could be used for both producing toy

components and assembling. One capacity hour can be used to produce $30$ sets of toy

components or to assemble $15$ units of toys. All the related costs for the manufacturing and

assembling are listed below. No other costs and tax are considered. The manufacturing

overhead cost is fixed and allocated to the units produced based on their production time. We

assume that the demand is high enough so that the manufacturing capacity is always utilized

fully.

The direct materials, direct labor and manufacturing overhead for each set of toy components

are $$4\mathrm{.}40,$$$2\mathrm{.}80$ and $$0\mathrm{.}80,$ respectively.

The additional direct materials, direct labor and manufacturing overhead for each unit of

assembly are $$2\mathrm{.}00,$$$1\mathrm{.}00$ and $$1\mathrm{.}60,$ respectively.

$(1)$ The XYZ executives decide to assemble the toys themselves instead of selling the

components. With the expected sale price of $$16$ per unit, is this a good decision?

$(2)$ What is the lowest price for the completely assembled $($finished$)$ toys that will be

acceptable for the executives' decision?

$(3)$ The manufacturing overhead cost in $(1)$ and $(2)$ is a fixed cost allocated to each unit.

Suppose that $50\%$ of that fixed overhead cost is variable. Please reevaluate your answers

in $(1)$ and $(2).$