Respuesta :
We have been given that Jimmy invests $17,000 in an account that pays 4.70% compounded quarterly. We are asked to find the time it will take for Jimmy's investment to reach $22,000.
We will use compound interest formula to solve our given problem.
[tex]A=P(1+\frac{r}{n})^{nt}[/tex], where,
A = Amount after t years,
P = Principal amount,
r = Annual interest rate in decimal form,
n = Number of times interest is compounded per year,
t = Time in years.
[tex]P = 17,000[/tex], [tex]A=22,000[/tex], [tex]n=4[/tex] and [tex]r=\frac{4.70}{100}=0.047[/tex].
[tex]22,000=17,000(1+\frac{0.047}{4})^{4\cdot t}[/tex]
[tex]22,000=17,000(1+0.01175)^{4\cdot t}[/tex]
[tex]22,000=17,000(1.01175)^{4\cdot t}[/tex]
Switch sides:
[tex]17,000(1.01175)^{4\cdot t}=22,000[/tex]
[tex]\frac{17,000(1.01175)^{4\cdot t}}{17,000}=\frac{22,000}{17,000}[/tex]
[tex](1.01175)^{4\cdot t}=\frac{22}{17}[/tex]
Now we will take natural log on both sides:
[tex]\text{ln}((1.01175)^{4\cdot t})=\text{ln}(\frac{22}{17})[/tex]
Applying rule [tex]\text{ln}(a^b)=b\cdot\text{ln}(a)[/tex], we will get:
[tex]4t\cdot\text{ln}(1.01175)=\text{ln}(\frac{22}{17})[/tex]
[tex]t=\frac{\text{ln}(\frac{22}{17})}{4\text{ln}(1.01175)}[/tex]
[tex]t=\frac{0.2578291093020998}{4\cdot 0.0116815047738379}[/tex]
[tex]t=\frac{0.2578291093020998}{0.0467260190953516}[/tex]
[tex]t=5.517891622[/tex]
0.52 years will be approximately 6 months.
Therefore, it will take 5 years and 6 months to reach Jimmy's investment to reach $22,000.
