Expert: Paul Klarreich - 5/20/2008

Question
Please help~

A manufacturer who sells to customer located within R km of his mill charges a per unit price m plus a shipping charge of s per kilometre. Thus a customer located x km from the mill would pay

P = m + sx          0 < or = x < or = R
per unit.
Our manufacturer now wants to know the average delivered price per unit. This clearly depends on the distance of each customer from the mill and how much each customer buys.

This information is summarised by a function f(x) that gives the distribution of sales by distance. In other words, f(x) gives the number of units sold to customers located x km from the mill.

(a) Using the simple idea that the average delivered price (A) is total revenue divided by total number of units sold for customers located within R km of the mill, derive an expression for A in terms of m, s, R and f(x).
(b) Calculate a numerical value for A for the case m=100, s=1, R=30 and f(x)= 900-x^2  

Answer
Questioner:   Fung wai wing
Category:  Advanced Math
Private:  No
 
Subject:  Application of Integration
Question:  Please help~

A manufacturer who sells to customer located within R km of his mill charges a per unit price m plus a shipping charge of s per kilometre. Thus a customer located x km from the mill would pay

P = m + sx          0 < or = x < or = R
per unit.
Our manufacturer now wants to know the average delivered price per unit. This clearly depends on the distance of each customer from the mill and how much each customer buys.

This information is summarised by a function f(x) that gives the distribution of sales by distance. In other words, f(x) gives the number of units sold to customers located x km from the mill.

(a) Using the simple idea that the average delivered price (A) is total revenue divided by total number of units sold for customers located within R km of the mill, derive an expression for A in terms of m, s, R and f(x).
(b) Calculate a numerical value for A for the case m=100, s=1, R=30 and f(x)= 900-x^2
..............................
Hi, Mr Fung,

I think you want to write: "a customer located x km from the mill would pay a shipping charge of  P = sx  per unit, and the number of units sold at x is  f(x)."

[Clearly the m is irrelevant.  You only care about the shipping charge.]

So you want the average value of the product:  cost/unit * units, divided by the number of units sold.

That would be:
    Integral[0 to R] of sx f(x)
A = -------------------------- PLUS m, of course.
    Integral[0 to R] of f(x)

For your example:

    INT(0..30) [x(900 - x^2)]
A =  -------------------------
    INT(0..30) [900 - x^2]

    INT(0..30) [(900x - x^3)]
A =  -------------------------
    INT(0..30) [900 - x^2]

    [(450x^2 - x^4/4)](0..30)
A =  -------------------------
      900x - x^3/3][0..30]

    [(450(30)^2 - (30)^4/4)]
A =  -------------------------
       900(30) - (30)^3/3

    [(450(30) - (30)^3/4)]
A =  -------------------------
        900 - (30)^2/3

    [(450 - (30)^2/4)]
A =  -------------------
        30 - (30)/3

    450 - 225
A =  ----------
        20  

    225
A =  --- = 45/4, plus 100, of course.
    20