Shum Manufacturing, which uses the high$-$low method for estimating cost function, makes a product called Kwan. The company incurs three different manufacturing cost types $($A$,$ B$,$ and C$)$ and has a relevant range of operation between $4,000$ units and $10,000$ units per month. Per$-$unit costs at two different activity levels for each cost type are presented below.

Type A Type B Type C Total

$5,000$ units $$7$ $$12\mathrm{.}60$ $$4$ $$23\mathrm{.}60$

$7,500$ units $$7$ $$8\mathrm{.}40$ $$3$ $$18\mathrm{.}40$

Required:

a$)$ Classify each of the costs $($A$,$ B$,$ and C$)$ as either fixed or variable or semi$-$variable

b$)$ Write down a linear cost function that expresses the behavior of Shum's total manufacturing cost as a function of # of units produced.

c$)$ If Shum produces $10,000$ units, what would be the total cost of manufacturing?

d$)$ If all fixed costs $($or fixed components of costs$)$ decrease by $10\%$ and all variable costs $($or variable components of costs$)$ decrease by $10\%,$ what would be the total cost of manufacturing for an activity level of production of $8,000$ units?