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Calculus/elemantary calculus

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Expert: - 2/11/2009


Question
Maximum Profit: 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.

Answer
Amount of money is M.

Interest rate is i and ki² = M, where k is a constant.

The bank can invest at 12%.

Profit = P(i) = 0.12M – iM, but we know M is related to i by M=ki².

Putting in M=ki², we get P(i) = 0.12ki² – ki^3.

Taking the derivative gives us DP/di = 0.24ki – 3ki² = k(0.24-3i).

From here, we need the derivative set to 0 to find the maximum,
so we have 0.24 - 3i = 0, which means i should be 0.08.

This surprises me a bit since I thought that k would be invovled,
but apparently its not.

Calculus

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