Interest rate that generates the maximum profit Story Problem:

A savings and loan association estimates that the amount of money on
deposit will be 1 million times the percentage rate of interest. For
instance, a 4% interest rate will generate $4 million in deposits. If
the savings and loan association can loan all the money it takes in at
10% interest, what interest rate on deposits generates the greatest
profit?

This is a problem that is part of a chapter where derivatives are used to solve business and economic problems.
The problem says to find the greatest profit. The profit function and associated functions/vars are

*

*Profit: P(x) = R(x) - C(x)

*number of items a business has sold = x

*Revenue: R(x) = x*p

*price of an item sold = p

*Cost: C(x) = cost of selling x

I'm having a hard time converting the info in the story problem to the profit formula.
It seems like:

*

*Revenue = 1million * interest rate

*Cost = ?

*

*I'm not sure what the cost is.  I don't see any indication of what the cost of a deposit is.



*Profit = (1,000,000 * i) - ?Cost?

Thank you for your help!
 A: Looking at revenue and cost for a bank is confusing because it's often backwards from how people or companies think of it.  The bank loses money by paying interest on deposits.  Why does it take deposits then?  So that it  can turn around and loan that money out to other people at a higher interest rate.
The bank makes money by collecting interest on loans.  The higher the interest on the loan, the more revenue.  So the revenue will be the total amount of money the bank is able to loan out times the loan interest rate. The bank cannot loan out more money than it has received in deposits.

*

*number of items a business has sold = dollars loaned

*price of an item  sold = loan interest rate

The bank loses money by paying interest on deposits.  That's how it convinces customers to use its bank.  So the cost is the total amount of money on deposit times the deposit interest rate.

*

*cost of selling 1 item = cost of loaning one dollar out = cost of convincing one person to deposit one dollar in the bank = deposit interest rate

This problem is extra confusing because both revenue and cost are (total money)*(interest rate).  You have to pay attention to whether it is the interest rate on loans the bank is offering, or the interest rate on deposits.
