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					                                   The Cookie Problem

       This semester my colleagues and I used a simple modeling problem as one of the

questions on our Precalculus final exam. The question was,

       "The Fresh Bakery currently sells 1000 large chocolate chip cookies each

       week for $0.50 each. The bakery would like to increase its revenue from

       the cookie sales by increasing this price. A survey convinced the manager

       that for every $0.10 increase, the bakery would sell 70 fewer cookies each

       week. At what price should the cookies be sold to maximize the revenue

       to the bakery?”

The Anticipated Solution

       The revenue, R, is the product of the price charged, p, and the number of cookies

sold, C. So, R  p  C . The number of cookies sold is a linear function of price, since for

each 10 cent increase in price, the demand is reduced by 70 cookies. The slope of the line

is

                                   70 cookies           cookies
                          m                       7               .
                                 10 cent increase      cent increase

                                                                                  
Since the bakery presently sells 1000 cookies at 50 cents each, the point 50,1000 lies on

the line. So

                                      
                               C  7 p  50  1000  7 p  1350

The revenue, then, is given by

                                        
                                      R p  p 7 p  350 .
This is a quadratic function and the optimum price is represented by the p-coordinate of

the vertex of this parabola.
                                                    4
                                              7.5 10




                      Revenue in cents
                                                      4
                                                  5 10

                                         R( p )
                                                    4
                                              2.5 10



                                                          0
                                                              0   40   80         120     160   200
                                                                             p
                                                                       P rice in cent s


We expected that students would find the vertex in one of three ways.

   Recall from Algebra the location of the vertex of the quadratic y  ax 2  bx  c is

         b                        1350
    x      , or in this case, p         96.43 .
                                         
         2a                         14

   Remember the geometry of a quadratic function with the vertex at the midpoint of the

    zeros, and recognize the given quadratic is factored, so the zeros are easily found at

                                                                       1350
                  1350                                                       0
    p  0 and p       . The vertex is at p                             7       96.43 .
                                                                                
                    7                                                      2

   Graph the function R on their calculators and seek the numerical maximum, which is

    reported as p  9643.
                   .

         Regardless of the technique, we expected to have the solution reported as the

optimum price is p  96 cents. We expected several students to consider the optional
                   

prices of 95 cents or $1.00, since these are more practical values and the difference in

revenue is negligible. We also thought we might find some solutions adjusting the price

so that, with tax, the total would be exactly $1.00.

         My colleagues and I expected some variability in the process used to locate the

vertex and in the numerical value presented as the solution, but we never expected the
variaty of approaches used in setting up the problem. In all, we had seven different

approaches to the modeling aspect of the problem.

Student Solutions

       Solution 1 was the anticipated solution, and was a fairly common solution path.

       Solution 2 is a modification of the anticipated solution.        The students first

developed a table of price and demand.

  Price        50      60       70     80      90     100    110    120
 Demand       1000    930      860    790     720     650    580    510


They then used their calculators to fit the regression line relating Price to Demand. This

                                                                             
line is y  7x  1350. Next, they created the revenue function R  x  y  x 7 x  1350 .
       Solutions 3 and 4 carried this data analytic approach one step farther. These

students multiplied the two rows in the table together and created a table with revenue.

 Price          50       60       70        80        90     100      110      120
Demand         1000     930      860       790       720     650      580      510
Revenue       50000    55800    60200     63200     64800   65000    63800    61200


       Some students simply looked on the chart and reported the optimum price as

$1.00 and the expected revenue as $650.00. A more common approach was to use the

calculator to fit a quadratic function to the data       Revenue using
                                                        price,                    quadratic

regression.
                                                    4
                                                7 10

                                                     4




                     Revenue in cents
                                             6.375 10

                                        R            4
                                            i 5.75 10

                                                     4
                                             5.125 10

                                                     4
                                               4.5 10
                                                         40      56     72           88        104   120
                                                                             P
                                                                                i
                                                                        P rice in cent s



This quadratic is y  7 x 2  1350x .                          From here they found the vertex of the quadratic

model.

         The first four solutions used price as the independent variable. Some students

used units of cents while others used dollars, but the solutions generally followed those

outlined above. If the independent variable is price, then the algebraic task for the

students is to write the demand function in terms of price. However, the other solutions

used different aspects of the problem as the independent variable.

         Solution 5 used the number of dime increases, d, as the independent variable. In

this case, the product of price and demand is more complicated to write. Both price and

demand must be written in terms of the number of dime increases. The price is written as

                                                                  
                                                               p  .50  01d ,
                                                                          .   
and the demand is written as

                                                                 
                                                              D  1000  70d ,   
so the revenue function is

                                                            .       
                                                                       
                                                  R d  .50  01d 1000  70 d .            
Students who located the vertex as the midpoint of the zeros used this form of R,

otherwise they re-wrote it as
                                      
                                     R d  7d 2  65d  500 .

The vertex of this parabola is at

                                                65
                                          d        4.64 .
                                                   
                                                14

           As you might suspect, there were some solutions suggesting the best price is 46

cents, but most students who got this far realized the solution represented 4.64 dime

increases for a price of 96 cents.

           Solution 6 was the most difficult of all the solutions. In this approach, the student

let the independent variable be the number of cookies sold. There were very few correct

solutions with this method. In this case, we need to write the price in terms of the number

of cookies sold rather than cookies in terms of price.                One correct solution used the

model

                                       
                                         
                                    R C  C 05 
                                             .
                                                     1000  C
                                                       700
                                                              .  
                                                                 
It was difficult to determine the student's reasoning in developing this model. Here's how

it goes:

                  “Since we lose 70 cookies with each 10 cent increase, we lose 700

           cookies for each dollar increase. The number of cookies lost, L, is 700

           times the increase in price measured in dollars, I, so

                                                               L
                                     L  700  I and I           .
                                                              700

                                           
           The price we charge is 0.50  I . If C is the number of cookies sold, then

           the number of cookies lost due to increasing the price is L  1000  C , and

           the price we charge is
                                                       1000  C      
                                                  
                                         price  05 
                                                  .
                                                          700
                                                                 .
                                                                      
       Putting this together generates the function

                                          
                                            
                                        R C  C 05 
                                                 .
                                                        1000  C
                                                          700
                                                                 .”   
                                                                      
       This is also a quadratic function.

                             700


                             525

                       R( b ) 350


                            175
                      Revenue in Dollars

                               0
                                    0       200       400       600        800   1000
                                                           b
                                                  Number of Cookies Sold


The vertex is located at C  635 . This means the best price to charge is

                                                 1000  635    
                               price  05 
                                        .
                                                    700        
                                                              0.96 .
                                                             


       The final solution was the most difficult to understand. Perhaps you will see

immediately what the student did. We certainly didn’t. As an exercise, work out the

student’s thought processes in developing this model. The model given for revenue was


                                          .0570.
                                            01. 
                                           
                                             x
                                    R x  x 1000 
                                                     
What is x, and what line of reasoning was used to create this particular function? Is this

solution better or worse than others given?

       My colleagues and I were surprised by the variety and creativeness used in solving

what I thought would be a relatively simple, straight-forward problem. The issues of

assessing student solutions is a difficult one. Do you think all of these solutions are
equally good? Should some receive more credit than others? Which solutions do you

think illustrate a greater understanding of principles of precalculus than others? I would

appreciate any thoughts and comments on your assessment of these solutions. Perhaps

we can create another article from your ideas on the assessment of these problems. Let us

know what you think. Send your ideas and suggestion by e-mail to:

compton@odie.ncssm.edu or teague@odie.ncssm.edu

or write to:

Helen Compton and Dan Teague

North Carolina School of Science and Mathematics

1219 Broad Street

Durham, North Carolina 27715

				
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