Expert: Paul Klarreich - 3/1/2007

Question
Hiya can you please help me with this question.i have a feeling what i have done is incorrect...



1. A music company sells music downloads they charge £1 per song and
average 1400 download per week. they run an experiment and notice that for
every 10 pence they decrease the price they sell 100 more songs.

(a)  draw a table that shows how the price decrease affects the price per song,
the numer of songs download and weekly income.
include rows of results for price decreases of 0,1,2,3 and x ten pence.


(b) Write down the quadractic equation which gives the total income for a
price decrease of x ten pence pieces.


(c) plot a graph of the total income, y against the decrease, x, in the number
of ten pence for values of x between -5 and 5 inclusive.



(d) Using your plot what should the company charge to get the maximum
daily income? Confirm numerically that this is the maximum by calculating
the income for x values on each side of your best value.  

Answer
Questioner:   pearl
Category:  Advanced Math
Private:  No
 
Subject:  word question
Question:  Hiya can you please help me with this question.i have a feeling what i have done is incorrect...

>> What have you done?  [she shrieked!]

1. A music company sells music downloads they charge £1 per song and
average 1400 download per week. they run an experiment and notice that for
every 10 pence they decrease the price they sell 100 more songs.

>> which is the same as saying for every decrease of  0.01 smackers in the price the increase in sales is 10.  OR for every decrease of 1, the increase is 1000.  Gotta have consistent units.

>> So  dS/dV = -1000

(a)  draw a table that shows how the price decrease affects the price per song,
the number of songs download and weekly income.
include rows of results for price decreases of 0,1,2,3 and x ten pence.

(b) Write down the quadractic equation which gives the total income for a
price decrease of x ten pence pieces.

(c) plot a graph of the total income, y against the decrease, x, in the number
of ten pence for values of x between -5 and 5 inclusive.

(d) Using your plot what should the company charge to get the maximum
daily income? Confirm numerically that this is the maximum by calculating
the income for x values on each side of your best value.
..............................
Hi, Pearl,

This is a max-min problem, and the general scheme is:

1. Find the quantity to be maximized.  In this case, it is the total income.  Give it a name:

Let  T = the total income.

2. Find the variable that is to be 'adjusted'.  There might be more than one.  In this case it is the price variance.  Give it a name:

Let  V = price Variance.

3. Find a way to express the quantity to maximize (or minimize) in terms of the variable.  This may involve some work.

In this example, the variance V, certainly affects the price, P (which we didn't give a name to, yet) in the obvious manner:

P = 1 + V

[I'm going to do the obvious thing, which is to consider that V is + if there is an increase.]

But the conditions say that the number of sales, S, depends on the Variance:

S = 1400 - 1000V

Check that out:  If the variance is 10 cents, [sorry, Pearl, I only use US currency here.] which is 0.10 whatevers, you get 100 increase.  That looks good.

Now we are ready for our Total income.

T = Price * Sales
T = (1400 - 1000V)(1 + V)
T = 1400 - 1000V + 1400V - 1000V^2
T = 1400 + 400V - 1000V^2
T = 200(7 + 2V - 5V^2)

OK, there's your quadratic function.  I think that the rest of it will be pretty routine.  This is a parabola which opens downward and has its vertex (a max) at

V = -b/2a = -2/2(-5) = 1/5 = 20 cents.  That's an increase, which means you should set the price at 1.20 smackers per 'song' for maximum revenue.

P.S. My spreadsheet gives this data:(USE COURIER FONT FOR THIS.)

Price   Sales   Revenue
0.4   2000   800
0.5   1900   950
0.6   1800   1080
0.7   1700   1190
0.8   1600   1280
0.9   1500   1350
1   1400   1400
1.1   1300   1430
1.2   1200   1440
1.3   1100   1430
1.4   1000   1400
1.5   900   1350
1.6   800   1280
1.7   700   1190