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#1 2016-06-08 19:14:27

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Making a profit function

Hello good people.

Today i was doing a word exercise and i was so happy because i thought that i finally managed to crack one of these on my own, but since that is impossible, when i checked the answers in the book, my answers were false. Would love to get some pointers on this exercise:

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The cost, in dollars, to produce x “I’d rather be a Sasquatch” T-Shirts is C(x) = 2x + 26, x ≥ 0
and the price-demand function, in dollars per shirt, is p(x) = 30 − 2x, 0 ≤ x ≤ 15.

* Find the profit function P (x).
* Find the number of items which need to be sold in
* Find the maximum profit.
* Find the price to charge per item in order to maxi
* Find and interpret break-even points.
-----------------------------------------------------------------------------------------------------------------------------------------

So to find the profit function, i thought that:

Profit(x) = Revenue(x) - Cost(x)

I thought i would set x to be the price, so that i could make the profit function out of the two functions provided in the exercise:

C(x) = 2x + 26, x here is the amount of items (am i interpreting this right?)
p(x) = 30 − 2x, x here is the price to find the amount of items that would sell (am i interpreting this right?)

I use the above to get:

R(x) = x(30-2x), revenue function where x is the price
C(x) = 2(30-2x)+36, cost function where x is the price

And then my profit function:

Profit(x) = (x(30-2x)) - (2(30-2x)+26),  where x is the price
Profit(x) = -2x^2 + 34x -96

Now with that profit function, to find the maximum profit at some price, i solve to get the vertex of the function:

Vertex formula = (-b/2a, f(-b/2a))
(17/2, Profit(17/2)) = (17/2, 48.5)

So i get that at a price of 8.5$ i maximize my profit at 48.5$. But of course that is wrong because the books says it should be (16$, 72$). Am i completely off the course here?

Last edited by sisyphus (2016-06-08 20:11:57)

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#2 2016-06-08 20:00:05

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Making a profit function


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#3 2016-06-08 20:26:32

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Re: Making a profit function

Thanks for the pointer, even though i don't understand it fully, i now know where i made the mistake. When i use 2x + 26 (instead of my 2[30-2x]+26) in the profit function, it all works out great, but i don't understand why do we use 2x + 26? i thought that the input to this function is the amount of items and the output is cost in dollars. Is that not the case? Or is it the other way around in that the input to the 30 - 2x function is the amount of items and the output selling price? I am not that bright, i know.

Last edited by sisyphus (2016-06-08 20:39:45)

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#4 2016-06-08 20:51:55

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Making a profit function

It is very simple. You have spent 2x+26 dollars to produce x shirts and you are selling them at a price of 30-2x dollars per shirt getting a revenue of x*(30-2x) , the excess of revenue over your expenditure is the profit. Please note that x is the number of shirts either produced or sold in the corresponding equations.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#5 2016-06-08 20:58:42

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Re: Making a profit function

thickhead wrote:

Please note that x is the number of shirts either produced or sold in the corresponding equations.

This is the part that i got wrong, i thought x was different things in those equations (dollars in one, items in another). It seems i am not very good at reading and understanding word exercises. Either way, thank you for your help yet again!

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#6 2016-06-09 00:39:31

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Making a profit function

I hope you are commerce/engineering student. It is always necessary to know practical aspects. In manufacturing there are 2 types of costs (1) fixed cost and 92) variable cost . Fixed cost does not depend on number of items produced like rent for premises, interest and depreciation on machinery,salary to people on permanent basis etc. The variable cost depends on number of goods produced like quantity of cloth used, threads for stitching etc.hence your cost is like 2x=36 as in the problem. Your selling price is decided by supply and demand. if you want to sell more goods you have to sell at lower price, like 30-2x as in problem. No mathematics does any good if it is not backed up by practical concepts.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#7 2016-06-09 16:07:45

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Making a profit function

I moght have erred at break even point. it is the minimum quantity you have to sell to  start making profit. So it should be only x=1 not 13.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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