Williams Inc. produces a single product, a part used in the manufacture of automobile transmissions. Known for its quality and performance, the part is sold to luxury auto manufacturers around the world. Because this is a quality product, Williams has some flexibility in pricing the part. The firm calculates the price using a variety of pricing methods and then chooses the final price based on that information and other strategic information. A summary of the key cost information follows. Williams expects to manufacture and sell $55,000$ parts in the coming year. While the demand for Williams$$s part has been growing in the past $2$ years, management is not only aware of the cyclical nature of the automobile industry, but also concerned about market share and profits during the industry$$s current downturn.

Total Costs

Variable manufacturing $ $4,670,000$

Variable selling and administrative $845,650$

Facility$-$level fixed overhead $2,335,875$

Fixed selling and administrative $665,495$

Batch$-$level fixed overhead $350,000$

Total investment in product line $22,340,000$

Expected sales $($units$)55,000$

Required:

$1.$ Determine the price for the part using a markup of $41\%$ of full manufacturing cost.

$2.$ Determine the price for the part using a markup of $21\%$ of full life$-$cycle cost.

$3.$ Determine the price for the part using a desired gross margin percentage to sales of $41\%.$

$4.$ Determine the price for the part using a desired life$-$cycle cost margin percentage to sales of $26\%.$

$5.$ Determine the price for the part using a desired before$-$tax return on investment of $15\%.$

$6.$ Determine the total contribution margin and total operating profit for each of the methods in requirements $1$ through $5.$

All asnwers must be given to $4$ decimal places.