question for math

Question

The marketing department of a local shoe company found that approximately

550

pairs of running shoes are sold monthly when the price of each pair is

$ 97

It was also observed​ that, for each

$ 1

reduction in​ price, an additional

30

pairs of running shoes are sold monthly. What price should the shoe store charge for a pair of running shoes in order to maximize​ revenue?

The shoe store charge should charge

​_____

for a pair of running shoes in order to maximize revenue.

Answer 1

Looking at the money taken in, the expressions would look like

Initial Case M = 550*97

One Reduction M = (550 + 30)(97 - 1)

Two Reductions M = (550 + 30 + 30)(97 - 1 - 1)

So,

Let E = the number of extra shoes sold, then express the Money (M) expression as

M = (550 + 30E)(97 - E) = 53,350 + 2360E -30E²

Now, taking the derivative dM/dE and setting this equal to zero, we have

2360 -60E = 0 or E = 39.33

This means that a reduction of $39.33 will give the maximum profit. But since the reduction is in $1 increments, you would do $39 or $40 & you will find that a $39 reduction is the maximum (You can check this!)

Therefore, The shoe price of $97 - $39 = $58 results in the maximum profit!