`r = 5%` Find the time necessary for $1000 to double when it is invested at a rate of r compounded (a) anually, (b) monthly, (c) daily, and (d) continuously

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Formula for compounding n times per year: `A=P(1+r/n)^(nt)`

Formula for compounding continuously: `A=Pe^(rt)`

A=Final Amount

P=Initial Amount

r=rate of investment expressed as a percent

n=number of compoundings per year

t=time in years

a) r=5% n=1 (annually)

`A=P(1+r/n)^(nt)`

`2000=1000(1+.05/1)^(1*t)`

`2=1.05^t`

`ln(2)=tln(1.05)`

`ln(2)/ln(1.05)=t`

`14.21=t`

Final Answer: 14.21 years

b) r=5% n=12 (monthly)

`A=P(1+r/n)^(nt)`

`2000=1000(1+.05/12)^(12*t)`

`2=(1.00416)^(12t)`

`ln(2)=12tln(1.00416)`

`ln(2)/[12ln(1.00416)]=t`

`13.89=t`

Final Answer: 13.89 years

c) r=5% n=365 (daily)

`A=P(1+r/n)^(nt)`

`2000=1000(1+.05/365)^(365*t)`

`2=(1.000136)^(365t)`

`ln(2)=365tln(1.00136)`

`ln(2)/[365ln(1.00136)]=t`

`13.86=t`

Final Answer: 13.86 years

d)`A=Pe^(rt)`

`2000=1000e^(.05*t)`

`2=e^(.05t)`

`ln(2)=.05tlne`

`ln(2)/[.05lne]=t`

`13.86=t`

Final Answer: 13.86 years

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial