Noah invested $890 in an account paying an interest rate of 4 5/8% compounded annually. Riley invested $890 in an account paying an interest rate of 5 1/4%
compounded continuously. To the nearest hundredth of a year, how much longer
would it take for Noah's money to triple than for Riley's money to triple?

3.37 years

Step-by-step explanation:

\text{Rate 1: }4\tfrac{5}{8}\%=4+5/8=

Rate 1: 4

8

5

​

%=4+5/8=

\,\,4.625\%\rightarrow 0.04625

4.625%→0.04625

\text{Rate 2: }5\tfrac{1}{4}\%=5+1/4=

Rate 2: 5

4

1

​

%=5+1/4=

\,\,5.25\%\rightarrow 0.0525

5.25%→0.0525

890\text{ tripled is }2670

890 tripled is 2670

\text{Calculate Tripling Time for Noah}

Calculate Tripling Time for Noah

\overline{\phantom{\text{Calculate Tripling Time for Noah}}}

Calculate Tripling Time for Noah

​

\text{Compounded Annually:}

Compounded Annually:

A=P(1+r)^t

A=P(1+r)

t

A=2670\hspace{35px}P=890\hspace{35px}r=0.04625

A=2670P=890r=0.04625

Given values

2670=

2670=

\,\,890(1+0.04625)^{t}

890(1+0.04625)

t

Plug in values

2670=

2670=

\,\,890(1.04625)^{t}

890(1.04625)

t

Add

\frac{2670}{890}=

890

2670

​

=

\,\,\frac{890(1.04625)^{t}}{890}

890

890(1.04625)

t

​

Divide by 890

3=

3=

\,\,1.04625^t

1.04625

t

\log\left(3\right)=

log(3)=

\,\,\log\left(1.04625^t\right)

log(1.04625

t

)

Take the log of both sides

\log\left(3\right)=

log(3)=

\,\,t\log\left(1.04625\right)

tlog(1.04625)

Bring exponent to the front

\frac{\log\left(3\right)}{\log\left(1.04625\right)}=

log(1.04625)

log(3)

​

=

\,\,\frac{t\log\left(1.04625\right)}{\log\left(1.04625\right)}

log(1.04625)

tlog(1.04625)

​

Divide both sides by log(1.04625)

24.2989=

24.2989=

Calculate Tripling Time for Riley

​

Compounded Continuously:

A=Pe^{rt}

A=Pe

rt

A=2670\hspace{35px}P=890\hspace{35px}r=0.0525

A=2670P=890r=0.0525

Given values

2670=

2670=

\,\,890e^{0.0525t}

890e

0.0525t

Plug in

\frac{2670}{890}=

890

2670

​

=

\,\,\frac{890e^{0.0525t}}{890}

890

890e

0.0525t

​

Divide by 890

3=

3=

\,\,e^{0.0525t}

e

0.0525t

\ln\left(3\right)=

ln(3)=

\,\,\ln\left(e^{0.0525t}\right)

ln(e

0.0525t

)

Take the natural log of both sides

\ln\left(3\right)=

ln(3)=

\,\,0.0525t

0.0525t

ln cancels the e

\frac{\ln\left(3\right)}{0.0525}=

0.0525

ln(3)

​

=

\,\,\frac{0.0525t}{0.0525}

0.0525

0.0525t

​

Divide by 0.0525

20.9259=

20.9259=

How much longer for Noah to triple:

24.2989-20.9259

24.2989−20.9259

3.373

3.373

20335+282=20617

step-by-step explanation: