Say Yes to the Dress! ...or, A Model MarriageName: ______Date: _____

1. Dress Sales

Mary Knupp-Shull runs a very posh (and therefore, very expensive) wedding dress boutique in Atlanta’s Buckhead neighborhood. A lot of people think that Mary’s life is all fabulous dresses and glamorous customers, but what makes Mary very successful is that she keeps a close watch on her sales and inventory.

Mary does not keep dresses in the store long, and she usually doesn’t do repeat orders on dresses because she likes to be on the cutting edge. The average time it takes for Mary to sell out of a dress is 13 months. The sales of one wedding dress model designed by Fabio Fabulisi is modeled by the function

,

where is the number of months since the release of the dress and is the number of dresses sold in multiples of ten.

(a) About how long did the dress stay on the shelves? How did the sales of this dress perform compared to the average dress model in Mary’s boutique?

(b) What was the most dresses sold in any one month?

2. Making Boxes

Mary needs open-topped boxes to store her excess inventory at year’s end. Mary purchases large rectangles of thick cardboard with a length of 78 inches and width of 42 inches to make the boxes. Mary is interested in maximizing the volume of the boxes and wants to know what size squares to cut out at each corner of the cardboard (which will allow the corners to be folded up to form the box) in order to do this.

(a) Volume is a three-dimensional measure. What is the third dimension that the value represents?

(b) Using the table below, choose five values of and find the corresponding volumes.

/ Length / Width / Volume

You tested several different values of above, and calculated five different volumes. There is a way to guarantee that you use dimensions that will maximize volume, and now we’re going to work through that process.

(c) Write an equation for volume in terms of the three dimensions of the box.

(d) Graph the equation from part (c).

(e) From your graph, what are the values of the three dimensions that maximize the volume of the box? What is the maximum volume of the box?

3. Mary’s Money

Mary has a whole team of bridal consultants who help customers pick out the perfect dress. Some customers find the perfect dress quickly, and some have to spend the entire day. Because this is such an involved process, Mary charges for the use of a consultant on an hourly scale. The first hour is free, and every hour after that is $25.00 per hour. A customer’s time is rounded up to the nearest whole hour.

(a) Graph the function that represents the fee structure for Mary’s bridal consultants.

(b) How much would a customer be charged if she stayed

(i) 59 minutes?

(ii) 61 minutes?

(iii) 180 minutes?

(iv) 493 minutes?

Mary’s tailoring department also has a fee schedule where labor for tailors is charged at $32.50 per hour, and the time taken on a dress is always rounded down to the next whole hour.

(c) Graphthe function that represents the fee structure for Mary’s tailors.

d) How much would a customer be charged if the tailoring for her dress took

(i) 30 minutes?

(ii) 60 minutes?

(iii) 119 minutes?

(iv) 121 minutes?

Mary has found that, to maximize her profits, she should sell 8 dresses per day for an expected daily profit of $1200. Each additional dress that she either sells or does not sell costs her $150 per dress. Mary will never schedule more than 12 appointments in a single day.

(e) What is the domain and range of the function that represents Mary’s daily profits?

(f) Graph the function that represents Mary’s daily profits.

(g) Is this function a polynomial? Why or why not?

(h) Write a single equation for this function.

(i) Write a piecewise equation for this function.

4. Daisy Mae’s Wedding

Daisy Mae has purchased a beautiful dress from Mary’s boutique and is now getting ready for her wedding day. Daisy Mae has figured out that the “sunk” cost of the wedding reception (including the dress, venue, permits, food and drink, etc.) is $24,000. Daisy Mae also knows that for every 30 minutes that her wedding reception lasts, the variable costs increase by $150.

(a) What is the cost per half-hour of Daisy Mae’s wedding reception if it lasts 3 hours?

(b) Develop a function that gives the cost per half-hour of the wedding reception as a function of the number of half-hours that the wedding reception lasts.

(c) Sketch a graph of this function. Give any asymptotes from the graph.

Vertical Asymptotes:

End-Behavior Asymptotes:

(d) Explain the meaning of the horizontal asymptote in terms of the wedding reception.