King Bathroom Fixtures $($KBF$)$ makes faucets, basins, and so on primarily for home use and sold through major retail chains. The design team at KBF has been working on a unique design to provide reasonable pressure while still conserving water. The market is quite competitive and KBF analysts believe that the fixture could sell for a unit price of $$50\mathrm{.}40.$

The cost accounting team at KBF has estimated the following manufacturing costs for the new design.

Direct materials $ $23\mathrm{.}25$

Direct labor $7\mathrm{.}30$

Manufacturing overhead $10\mathrm{.}95$

Total $ $41\mathrm{.}50$

An operating profit of $12$ percent of manufacturing costs is required for all new products at KBF without the explicit consent of the top executive team. At KBF$,$ operating margin is defined as revenues less manufacturing costs, all divided by manufacturing costs$).$

Required:

a$.$ Suppose KBF uses cost$-$plus pricing, setting the price equal to manufacturing costs plus $12$ percent of manufacturing costs. What price should it charge for the fixture?

b$.$ Suppose KBF uses target costing. What is the highest acceptable manufacturing cost at which KBF would be willing to produce the fixture?