Problem $4-60($Algo$)$ Target Costing $($LO $4-3)$

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 $$46\mathrm{.}40.$

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

$$

Direct materials$ $22\mathrm{.}25$Direct labor$6\mathrm{.}90$Manufacturing overhead$10\mathrm{.}35$Total$ $39\mathrm{.}50$

$$

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?