The new CEO of the company. Meredith feels a great responsibility to maintain the family

heritage with the company. She realizes that the company needs

to continue to develop and market exciting new products. Since

the $50$ th anniversary of the founding of the company is rapidly

approaching, she has decided to select a particularly special new

product to launch with great fanfare on this anniversary. But

what should it be$?$ As she ponders this crucial decision, Mer$-$

edith's thoughts go back to the magnificent grandfather clock

that her grandparents had in their home many years ago. She had

admired the majesty of that clock as a child. How about launch$-$

ing a modern version of this clock?

This is a difficult decision. Meredith realizes that grandfa$-$

ther clocks now are largely out of style. However, if she is so

nostalgic about the memory of the grandfather clock in her

grandparents' home, wouldn't there be a considerable number

of other relatively wealthy couples with similar memories who

would welcome the prestige of adding the grandeur of a beauti$-$

fully designed limited$-$edition grandfather clock in their home?

Maybe. This also would highlight the heritage and continuity of

the company. It all depends on whether there would be enough

sales potential to make this a profitable product.

Meredith had an excellent management science course as part

of her MBA program in college, so she realizes that break$-$even

analysis is needed to help make this decision. With this in mind,

she instructs several staff members to investigate this prospective

product further, including developing estimates of the related

costs and revenues as well as forecasting the potential sales.

One month later, the preliminary estimates of the relevant

financial figures come back. The cost of designing the grand$-$

father clock and then setting up the production facilities to

produce this product would be approximately $$250,000.$ There

would be only one production run for this limited$-$edition

grandfather clock. The additional cost for each clock produced

would be roughly $$2,000.$ The marketing department estimates

that their price for selling the clocks can be successfully set at

about $$4,500$ apiece, but a firm forecast of how many clocks can

be sold at this price has not yet been obtained. However, it is

believed that the sales likely would reach into three digits.

Meredith wants all these numbers pinned down considerably

further. However, she feels that some analysis can be done now

to draw preliminary conclusions.

a$.$ Assuming that all clocks produced are sold, develop a spread$-$

sheet model for estimating the profit or loss from producing

any particular number of clocks.

b$.$ Use this spreadsheet to find the break$-$even point by trial and

error.

c$.$ Develop the corresponding mathematical expression for the

estimated profit in terms of the number of clocks produced.

d$.$ Use a graphical procedure to find the break$-$even point.

$e.$ Use the algebraic procedure to find the break$-$even point.

A fairly reliable forecast now has been obtained indicating

that the company would be able to sell $300$ of the limited$-$edition

grandfather clocks, which appears to be enough to justify intro$-$

ducing this new product. However, Meredith is concerned that

this conclusion might change if more accurate estimates were

available for the various costs and revenues. Therefore, she

wants what$-$if analysis done on these estimates. Perform what$-$

if analysis by independently investigating each of the following

questions.

$f.$ How large can the cost of designing this product and setting

up the production facilities be before the grandfather clocks

cease to be profitable?

g$.$ How large can the production cost for each additional clock

be before the grandfather clocks cease to be profitable?

h$.$ If both of the costs identified in parts $f$ and $g$ were $50\%$ larger

than their initial estimates, would producing and selling the

grandfather clocks still be profitable?

i$.$ How small can the price for selling each clock be before the

grandfather clocks cease to be profitable?

Now suppose that $300$ grandfather clocks are produced but

only $200$ are sold.

j$.$ Would it still be profitable to produce and sell the grandfather