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 $4000 is deposited in a savings account drawing 5 3 4 % 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 = 4000 1 + 1 4 (0.0575) 5(4) that is accrued when interest is compounded quarterly. (Round your answer to the nearest cent.) S = $

Answer 1

Answer:

a) - r=5%: $S= 5,136.10$

- r=4%: $S= 4,885.61$

- r=3%: $S= 4,647.34$

b) - r=5%: t=14 years

- r=4%: t=17 years  [/tex]

- r=3%: t=23 years  [/tex]

c) The amount obtained is

- Compuonded quarterly: $5,191.83

- Compuonded continously: $5,200.71

The latter is always greater, since the more often it is capitalized, the greater the effect of compound interest and the greater the capital that ends up accumulating.

Explanation:

The rate of accumulation of money is

$dS/dt=rS$

To calculate the amount of money accumulted in a period, we have to rearrange and integrate:

$\int dS/S=\int rdt=r \int dt\\\\ln(S)=C*r*t\\\\S=C*e^{rt}$

When t=0, S=S₀ (the initial capital).

$S=S_0=Ce^{r*0}=Ce^0=C\\\\C=S_0$

Now we have the equation for the capital in function of time:

$S=S_0e^{rt}$

a) For an initial capital of $4000 and for a period of five years, the amount of capital accumulated for this interest rates is:

- r=5%: $S=4000e^{0.05*5}=4000*e^{0.25}= 5,136.10$

- r=4%: $S=4000e^{0.04*5}=4000*e^{0.20}= 4,885.61$

- r=3%: $S=4000e^{0.03*5}=4000*e^{0.15}= 4,647.34$

b) We can express this as

$S=S_0e^{rt}\\\\2S_0=S_0e^{rt}\\\\2=e^{rt}\\\\ln(2)=rt\\\\t=ln(2)/r$

- r=5%: $t=ln(2)/0.05=14$

- r=4%: $t=ln(2)/0.04=17$

- r=3%: $t=ln(2)/0.03= 23$

c) When the interest is compuonded quarterly, the anual period is divided by 4. In 5 years, there are 4*5=20 periods of capitalization. The annual rate r=0.0525 to calculate the interest is also divided by 4:

$S = 4000 (1+(1/4)(0.0525))^{5*4}=4000(1.013125)^{20}\\\\S=4000*1.297958= 5,191.83$

If compuonded continously, we have:

$S=S_0e^{rt}=4000*e^{0.0525*5}=4000*1.3= 5,200.71$

The amount obtained is

- Compuonded quarterly: $5,191.83

- Compuonded continously: $5,200.71

The latter is always greater, since the more often it is capitalized, the greater the effect of compound interest and the greater the capital that ends up accumulating.