Average rates of change question

Question

A shopkeeper calculates that for each reduction of $1 in the price of a $50 pair of jeans, another customer will buy a pair. Show that if 10 customers will buy the jeans at full price, sales receipts will be S=(10+x)(50-x), where x is the price reduction in dollars. The jeans cost $24 per pair.

a) Find an expression for the profit, P, made on jeans sales in terms of x

b) Find dP/dx and discuss the meaning of the situation dP/dx=0

c) Find for what value(s) of x dP/dx=0. What advice would you offer to the shopkeeper?

 

I'm super stuck on part a! 

I know the profit will need to be the total amount earned through sales, but you will need to take away the money lost in reducing the price I assume?? I have no idea how to find these figures :/

 

 

Answer 1

a)  They are asking for the profit (profit = sales - costs).

      sales = S = (10+x)(50-x),  where x is the price reduction in dollars

 

 In the eqn for S notice that  (10+x) =(number of jeans sold) and

        (50-x) = (price of a pair of jeans),  so the price reduction is built in.

 

    costs = C = 24(10+x)  

 

    profit = P = S - C = (10+x)(50-x) - 24(10+x)

     

    Simplify P ...

 

             P = (10+x)(50-x-24)

               ...

            

 

I hope this helps.   Remember that you are finding the profit function P as it relates to the discount offered on the jeans.  Let me know if you have any questions.