If you borrow P₀ dollars form a bank at an interest rate r (compounded continuously) and pay back the loan continuously at a rate of N dollars per year then the amount still owing to the bank, P(t), where t is measured in years satisfies the differential equation

P (t) = rP(t) N, P(0) = P₀.

Suppose you have a loan of $10,000 and the interest rate is 10% (r = 0.1).

(1) Find P(t) if N = 1200.

(2) How long will it take to pay back the loan in this case?

(3) If N = 800 will the loan ever be paid off? Explain.

(4) How large should N be to guarantee that the loan will be paid off in 10 years?