CX Republic Wheel Builders $($CX Republic$)$ provides hand$-$built wheels for the cyclocross races in the Salt Lake City, UT area. CX Republic operates out of a $1,200$ square foot space in an industrial park. Rent for the space is $$800$ a month and includes all utilities. CX Republic is a closely held operation that maintains one employee in addition to the owner. The employee is paid a salary for $$1,040$ a month plus $$20$ for each wheel built. The owner $($you$)$ is a residual claimant on the business, which means that you are classified as an investor, not an employee of the business. Because you believe Chris King provides the best hubs, Mavic provides the best rims, and SAPIM provides the best spokes, you only build wheels with these components. The cost of a Mavic reflex rim is $$40,$ a front wheel Chris King hub is $$100,$ a Chris King rear hub is $$210,$ and SAPIM spokes cost $$10$ per wheel. Your wheel builder uses a tension meter, a wheel stand, and some tools to build the wheels. These tools cost $$1,200$ and are expected to last three years. There are also administrative costs associated with running the business that you expect to amount to $$2,000$ for the year. Finally you will spend $$300$ a month on advertising expenses. Note that a set of wheels includes one front and one rear wheel. You have four tasks:

$(1)$Determine the cost of producing a set of wheels.

$(2)$Suggest a sale price for a set of wheels $($front and rear$).$

$(3)$Determine the predicted profit for the first quarter of this division if the firm produces and sells $100$ sets of wheels in the quarter where the sale price is set to $$800.($Assume no taxes and all costs are as described above$).$

$(4)$Determine the predicted profit for the first quarter of this division if the firm produces and sells $200$ sets of wheels in the quarter where the sale price is set to $$800.($Assume no taxes and all costs are as described above$).$