Jordan Martin
Franklin & Marshall College
Lancaster, PA 17604
February 1, 2008
William Avering
Blezz Shoes
Beckinforth, PU 22222
Dear Mr. Avering:
Based on my interpretation of the data that you provided, you had good reason to be unsure about the wisdom in selling your rubber back to that Vellen when you did. If you had sold back the machine and all the rubber you made on January 15th, 2008, instead of January 31st, you would have made over $100,000 dollars more, and had a two week vacation to boot. Let me explainÉ
There was a lot of information in that letter you wrote me, and I know you wrote it a while ago, so I want to summarize the important stuff. The machine Eve L. Vellen sold you could be bought back at any time for the same price, so we arenÕt concerned with that. The main focus was the amount of rubber Blezz Shoes produced (in buckets per day), and how that value changed over the course of time. We also need to take into account the variable price of rubber, in dollars per bucket.
Fortunately, the two graphs you gave me told me exactly what I needed to know. I included them in this letter so you could reference them if you needed to. The first thing we want to look at is the graph with the price of rubber (fig. 1). On the 360th day of the year, the rubber market was booming. One bucket was worth a little over $36. This was the peak price you could have sold your rubber for all year. You also had made more total rubber on this day than any other day before it. So, youÕd be selling more rubber than ever before for a higher price than it would be possible to sell for prior to that date. I concluded that it could not have been more profitable to sell on any day before the 360th (which happened to be December 26th). Apparently rubber is not a big hit as a Christmas present, or the data would have looked a little different. What about after the 360th day?
At this point, we need to start looking at the other graph (fig. 2), and we have to use numbers instead of just logic; no worries though, itÕs nothing complicated! This graph shows us how many buckets per day your rubber machine produced from January 1st, 2007 until the day you sold it back. The graph doesnÕt have the specific data points marked, and we donÕt have a formula to figure out exactly how many buckets were produced each day. If we focus on a small piece of the graph at a time Ð letÕs use ten day increments, since those are marked Ð we can find the average amount of rubber produced over the course of ten days. If we add all those ten day pieces up, weÕll end up with an estimate of the total amount of rubber you made, and therefore what you sold. The area of a rectangle is recognized as:
Area = Height x Width.
So, hereÕs what the calculation for one ten day increment looks like:
10 days x (40 buckets/day) = 400 buckets over a 10 day period.
You may be wondering where the 40 came fromÉ hereÕs why. There are a few different ways to find the area of each 10 day increment. You could start with the point (0,0) and consider the area to the right of this point. This would give your first rectangle an area of 0, because there is no height to multiply by. Your second rectangle would begin at (10,0), its width 10, its height 40, and its area would be 400 (the graph appears to show 40 buckets per day on day 10). Or, you could start with the point (10,0) and consider the area under the graph to the left of this point. This gives your first rectangle an area of 400, if you agree that on day 10, 40 buckets were produced per day.
This method was the one that I chose. For each new rectangle, I evaluated the graph at (x,0) and (x,y). Each y value was then multiplied by 10, to account for the 10 days of rubber production represented by the area. I included the graph with my rectangles shaded (fig. 3) so that you could better visualize what I did, and a data table showing my estimate of the amount of buckets per day every ten days (table 1). We must assume that any disparity that might arise between the actual amounts of buckets produced and my estimate would be slight, because this method overestimated some ten day periods, and underestimated others.
Now
that we know
approximately how many buckets were produced by any day that is a
multiple of
10, up until day 397 when you sold the rubber (we must use the bucket
amount on
day 400 for this), we can determine which day would have been the best
choice
to sell. We do this by multiplying the amount of buckets produced by
the price
in dollars per bucket, to find the total amount in dollars which would
be owed
to you by Vellen. I have included another data table (table 2) that
shows the
day, the amount of rubber to be sold on that day, the price per bucket,
and the
total amount you would have received had you sold on that day,
Because the most profitable date and the one that you actually sold on are not the same, I am sure that you are, quite understandably and rather regrettably, distressed. As this state does not favor the clearest thinking, I will recap my answer for you. You wanted to know when I thought you should have sold the machine (for $567,234, no matter what the date) and all the rubber (a variable amount, depending on the date). You also wanted to know if selling earlier or later would have made a difference, and if so, how much of a difference. Had you sold on the 380th day, January 15th, you would have made $113,120 more. The total amount of money you would have made including the machine would have been $2,033,950.
Okay Mr. Avering. This is my advice: go ahead and take that vacation anyway. After news like this, you deserve it. If at the end of your vacation, preferably tropical, you feel more relaxed or even somewhat charitable, think about writing a letter to the following people to thank them. The lovely ladies of Marshall 219, Claire and Tanya, helped me get started on this problem. Mobeen, my partner in crime (and Wednesday homework assignments) made me feel better when I called him after 12 AM and he was still writing to you too! Of course, we cannot forget the creative and very cool Dr. Crannell. SheÕs helping me wake up my memories of calculus, which have been dormant for a while. Last but not least, I must thank the vending machine that provided the chocolate bar that functioned as the carrot on a stick.
Best wishes in the shoe industry,
Jordan Martin
Calculus student
Franklin & Marshall College
Figure 1. Price of rubber in dollars per bucket, graphed against day of the year, 2007.
Figure 2. Output of the rubber machine, in buckets/day, graphed against day of the year, 2007.
Figure 3. Output of the rubber machine, in buckets/day, graphed against day of the year, 2007. The rectangles used to calculate bucket amounts are shaded.
Table 1. Average amount and total cumulative amount of rubber produced, days 10 Ð 400.
Table 2. Total rubber production, price/bucket of rubber, total profit, days 360 Ð 400.
look at the