Answer:
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
