word problem minimize profit

Question

An artisanal coffee shop finds that it can sell 500 cups of "hyper-local" coffee for $3 but that for every $0.25 more they charge, the number of cups they sell decreases by 25. What price should the coffee shop sell their coffee for in order to maximize their profits?

Answer 1

Using the naive approach, it is safe to say leave the price at $3.

More data is needed about the supply and demand, but based on what's given,

leave the price at $3. You already know the demand decreases once the price is raised

just one quarter.

 

Let's make a table:

 

Supply: vendor sells 500 cups of the coffee at $3 each.

 

  price    supply

--------------------

   $0           0

   $3       500

   

The equation of this line is y = 500X/3

 

Demand:

-----------

 

 Price          Demand

-------        --------------

$3                500

3.25            475

3.50            450

3.75            425

$4              400     

etc              etc     etc   as stated in the problem, the sales (demand)

                              decreases by 25 every time the price increases by 25 cents

 

The equation of this line is:   y = 800 - 100 x 

 

The equilibrium price is where these two lines meet:

 

500/3 X = 800 - 100 X

 

Solving for X:

 

 800/3 X = 800

 

 800 X = 2400

 

    X = 3

 

Leave the price at $3 and stock up for 500 cups of the coffee. 

They should sell steady at that price.