differential equations

Question

When interest is compounded continuously, the amount of money increases at a rate proportional to the amount S present at time t, that is, dS/dt = rS,

 where r is the annual rate of interest.

(a)

Find the amount of money accrued at the end of 5 years when $3000 is deposited in a savings account drawing 5.25% annual interest compounded continuously. (Round your answer to the nearest cent.)

$________________

(b)

In how many years will the initial sum deposited have doubled? (Round your answer to the nearest year.)

_________________ years

(c)

Use a calculator to compare the amount obtained in part (a) with the amount S =3000(1+(1/4)(0.0525))^(5(4)) that is accrued when interest is compounded quarterly. (Round your answer to the nearest cent.)

S = $ ________________

Answer 1

dS/dt = rS

(1/S)dS = rdt

∫(1/S)dS = ∫rdt

ln(S) = rt + C

S = e^rt+C = e^Ce^rt = Ae^rt

(a) S = 3000e^0.0525(5)

(b) 6000 = 3000e^0.0525t