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

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.

Step-by-step 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.