SOLUTION: Vinnie owns a t-shirt shop. He has determined that he can sell 150 t-shirts for $10 each. Vinnie's partner, Bob, is not happy with charging $10 for the shirts. He tells Vinnie,

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Question 1092693: Vinnie owns a t-shirt shop. He has determined that he can sell 150 t-shirts for $10 each. Vinnie's partner, Bob, is not happy with charging $10 for the shirts.
He tells Vinnie, "If we can increase the price by just one dollar we can make more money and maximize our revenue!" Vinnie responds, "But Bob, my research shows that each time we increase the price by one dollar, we will sell 5 less t-shirts! We will lose money!" Bob insists that they will make more money by charging more and selling fewer shirts.
Question:
How much should Vinnie charge in order to maximize the amount of money he makes?
Found 2 solutions by richwmiller, ikleyn:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
Vinnie should sell the shirts as he sees fit. It is his shop.
So he should sell the shirts for $10 each.
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.
For examples/samples/templates and guiding instructions on how to solve such problems see the lesson
    - Using quadratic functions to solve problems on maximizing revenue/profit
in this site.  You may also want to see the associated lessons with it by following the references of this lesson.


Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.


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It is deep night now, and I have time and opportunity to write the entire solution. 

So, what we have and what we do know from the condition ?
We know two very important things.


    1.  At the price $10 the store is determined to have 150 customers.

    2.  For every one dollar increase the price the number of customers decreases by 5.


In other words, having the price (10+x) dollars, they will have (150-5x) customers.


Then their revenue R is this function of the price:

R(x) = (150-5x)*(10+x)    (1)    (the number of customers multipled by the price)

And we are going to find the maximum of this quadratic function.


As this quadratic function is presented by the product of linear factors, you can easily determine its roots. They are 

    x =  = 30       (from the first factor)   and

    x = -10             (from the second factor).

The maximum of the quadratic function is achieved at the midpoint between the roots, i.e. at x=  =  = 10. 


It means that the optimal price for the shirt is 10+x = 10+10 = 20 dollars.

Selling at this price, they will have the number of customers  (150-5x) = 150-5*10 = 150-50 = 100,

and they will have the revenue  100*20 = 2000 dollars.

Compare it with those 150*10 = 1500 dollars and estimate the difference !

So, I solved this minimax problem for you, explained its meaning and the method of solution.


                    H a p p y   l e a r n i n g  ! !



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