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irina1246 [14]
3 years ago
9

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 = $
Business
1 answer:
liubo4ka [24]3 years ago
5 0

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.

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