Lusaka Mall Lttd has commenced business in Mandevu to meet the demand for assembling flat oack furniture that is being retailed by Tip$-$Top Ltd$.$L usaka Mall will employ its own trade$-$staff on a contract basis of which there is no shortage due to the current economic downturn.The disposable income for most households has been negatively affected because of reduced productivity due to the outbreak of pandemic.Lusaka Mall bases prices on the time for assembly plus any associated expenses.In the first month of operation staff reported that they are spending approximately $80\%$ of their hours assembling three standard products, wardrobe,a corner sideboard and kitchen units.Further reseach reveals that these are three of the most popular products.More worrying for the owner of Lusaka Mall are customer complaints,especially in the month,that they have been charged more than friends for assembly of the same product.A staff member has explaint this is probably due to the learnig effect.An analysis of job cards reveals the following information.

Cumulative Wardrobe Sideboard kitchen

Assembly Cumulative time Cumulative time cumulative time

$18\mathrm{.}0$ hours $4\mathrm{.}0$hours $20$ hours

$212\mathrm{.}8$ hours $7\mathrm{.}6$ hours $40$ hours

$420\mathrm{.}48$ hours $12\mathrm{.}96$ hours $80$ hours

$840\mathrm{.}96$ hours $23\mathrm{.}33$ hours $160$ hours

Trade staff is paid K$15$ per hour or part of and Lusaka Mall charges its customers K$25$ per hour.Lusaka Mall wishes to set standard selling prices for these products to avoid customer complaints.

Required:

$($a$)$ Calculate the rate of learning associated with the assemly of each product.

$($b$)$Calculate the selling selling price and profit per wardrobe for the fifth units onwards.

$($c$)$ If learning for sideboard ceases after $8$ units, calculate the selling price and profit per unit for every unit thereafter.