SOLUTION: A cereal company finds that the number of people who will buy one of its products in the first month that it is introduced is linearly related to the amount of money it spends on a

In this problem, the question/goal/function can be formulate two-fold, or in one of the two ways:


    (1)  for given number of boxes to sell, how much for advertising should be spend ?


    (2)  for given amount spent for advertising, how many boxes is expected to sell ?


            +------------------------------------------------+
            |   Which equation is requested in part (a) ?    |
            +------------------------------------------------+



They say  "Write a linear equation that relates the amount A spent on advertising 
           to the number x of boxes the company aims to sell".


So, they request an equation to be A = mx + b, where A is the amount to spent  for advertising
                                                 and x is the number of boxes.


Therefore, if you want to follow literally their request, you should use these points


          (100000,40000)  and  (200000,60000).   <<<---=== not that you wrote in your post.
                                                           

            +-------------------------------------------------+
            |        not that you wrote in your post.         |
            +-------------------------------------------------+



In other words, you should turn the pairs of numbers in your post "inside-out", 
as if you turn a usual sock inside out.  You should SWAP the numbers in your pairs.


After that, you may apply the standard formal procedure of making linear interpolation-extrapolation
function based on two points.


              +-------------------------------------------+
              |    But there is informal way of making    |
              |    this equation, which you may prefer.   |
              +-------------------------------------------+


From the given data, you see that for increment of the argument  x  of  200000-100000 = 100000,
you have an increment for the relevant function A(x)  of  60000-40000 = 20000.


                    the increment of A(x)       20000     2     1
So, the slope is   ------------------------ = -------- = --- = --- = 0.2.
                     the increment of x        100000     10    5


In other words, the company spends 0.2 of a dollar to increase the quantity of sold boxes by 1 box.


So, for now, your equation is  A(x) = 0.2x + b.


To find "b", substitute 40000 for A and 100000 for x into this equation.  You will get

    40000 = 0.2*100000 + b,

    40000 = 20000 + b,

    b = 40000 - 20000 = 20000.


So, finally the requested equation is 

 
    A(x) = 0.2*x + 20000,    (1)  


where x is the number of boxes expected to sell and A is the amount of dollars to spend for advertising.


You may CHECK:  at x= 100000 boxes, A(100000) = 0.2*100000 + 20000 = 20000 + 20000 = 40000  dollars ! correct !

                at x= 200000 boxes, A(200000) = 0.2*200000 + 20000 = 40000 + 20000 = 60000  dollars ! correct !


At this point,  part (a) is complete in full, as it is requested.




(b)  To answer question (b), you should evaluate  A(300000), using formula (1).




(c)  The interpretation of the slope in equation (1) is THIS: 

         the company spends 0.2 of a dollar to increase the quantity of sold boxes by 1 box.


         In other words, 0.2 dollars per box is the rate  
         "dollars spent for advertising per each additional box sold".

     This problem teaches us to TWO things:

          
          (a)  To read the problem attentively.

          (b)  To make the relevant Math in accordance with the problem's request.


     Both these components are critically important.