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Calculus/maximizing revenue


QUESTION: The sale of compact disks of "lesser" performers is very densitive to price.  If a CD manufacturer charges p(x) dollars per CD where p(x)=12-(1/8)x then x thousand CDs will be sold.  Find the maximum revenue.

I get the expression for the total revenue from the sale of x thousand CDs to be R=12000x-125(x^2)
I took the derivative R'= 12000-250x
I set that equal to zero to get x=48
When I put that in the equation for revenue, I get 0.  Where did I go wrong?

ANSWER: The equation for p(x) = (12 - x/8)x when x is in 1,000's of CD.
That is the same as p(x) = 12x - (x^2)/8.

Differentiating this says that p'(x) = 12 - x/4.
Setting this to 0 gives us x = 3.
Since this is in 1000's sold, that is 3,000 that will be sold.

Putting this into the equation gives p(3) = 12 - 3/8 = 11 5/8.
Multiplying this gives 3 * 11 5/8 = 33 15/8 = 34 7/8.
The answer is in 1,000's.

Since the equation has an x and and x^2 in it, multiplying the equation by 1,000 won't work.

---------- FOLLOW-UP ----------

QUESTION: Why did you change the original equation?  The equation was
p(x)= 12 - (x/8)  There wasn't an x with the 12 and the parentheses were with the second part of the equation.

ANSWER: I put that in because it is assumed to be there as it is written.

As written, p(x)= 12 - x/8 is really p(x)= 12 - (x/8), not p(x)= (12-x)/8,
for they give different values.  The reason I write it as p(x)= 12 - (x/8)
is to point out that division is done before subtraction.

According to basic mathematics, what is inside the parenthesis is done first.
As far as what's inside, do the powers first, multiplication and division next,
and then the addition and subtraction.  They are done from left to right.

For example, if x = 4+3*5, the way it is suppose to be done gives x = 4+15, so x=19.
The way that some people might do it would be to say x = 7*5 = 35, but this is incorrect.

That is one of the problems with putting equations on the computer.
They might be on two separate lines, but on the computer, they need to be on the same line.
I have seen some people write (a+b)/(c-d) as
but that only works if it is at the start of the line.

If it was in the question with the 8 beneath the 12-x,
then it should be written really p(x) = (12-x)/8.

My question to you, then, is this: how was it written?

---------- FOLLOW-UP ----------

QUESTION: Sorry.  I understand your confusion.  The problem is written
p(x)= 12 - (x/8)

The answer was said to be x=48, and I went back to the start to look at this.

The initial equation was looked at again it was found not to be 0.  p(48) is not 0, but 6.
See, if we put 48 in the equation, we get p(x) = 12 - 48/8 = 12 - 6 = 6.

In most problems of this nature, the maximum is half the value of the constant.
As can be seen, half of 12 is 6 and that is the maximum value.


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