Maximum Profit/Optimization problem (interest)

[AGlas9837]

I'm having a hard time getting started (coming up with primary equation) with the following: Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on the money. Furthermore, the bank can reinvest the money at 12% simple interest. Find the interest rate the bank should pay to maximize its profit.

I'm thinking the equation is something like P/x^2+12% but am sure I'm horribly wrong. It seems there is not enough information given for the problem.

 

[stapel]

I'm thinking the equation is something like P/x^2+12%....

What is "x"? What is "P"? What is your reasoning? You started with the proportionality equation you learned back in algebra, and... then what?

Please be complete. Thank you!

Eliz.

 

[AGlas9837]

I was using x for the interest rate and P for profit but after looking up the proportionality equation, I would guess my equation might look like: A=kB^2 where A is the amount of money deposited and B is the square of the interest rate. Am I getting anywhere?

 

[stapel]

...I would guess my equation might look like: A=kB^2 where A is the amount of money deposited and B is the square of the interest rate.

Help yourself by using the standard variables for principal, interest, and interest rate: P, I (capital "eye"), and r. So the given relationships are:

. . . . .amount deposited: P = kr[sup:37sip21c]2[/sup:37sip21c]

. . . . .amount earned: I = 0.12P

. . . . .amount paid: Pr

The profit is the difference between what the bank earns and what the bank pays. So substitute (to get rid of "P") and subtract (to create an expression for the profit p as a function of the rate r). Differentiate, and solve.

Eliz.