The nearest time period for Zachary to reach the amount is 14 years.
Compound interest is the interest on a deposit or loan calculated based on the initial principle, and the collective interest from previous periods.
For the given situation,
Principle, [tex]p=\$700[/tex]
Interest rate, [tex]r=4.9\%[/tex][tex]=0.049[/tex]
Amount, [tex]a=\$1360[/tex]
Compounded continuously, [tex]n=1[/tex]
We have to find the time period, [tex]t[/tex]
The amount formula of compound interest, [tex]a=p(1+\frac{r}{n} )^{nt}[/tex]
On substituting the values,
[tex]1360=700(1+\frac{0.049}{1} )^{1t}[/tex]
Divide by [tex]700[/tex] on both sides,
⇒[tex]\frac{1360}{700}=\frac{700}{700} (1+0.049 )^{t}[/tex]
⇒[tex]1.94=(1.049)^{t}[/tex]
Taking log on both sides,
⇒㏒[tex]1.94[/tex]=[tex]t[/tex]㏒[tex]1.049[/tex]
⇒[tex]0.28=0.020t[/tex]
Divide by [tex]0.020[/tex] on both sides,
⇒[tex]\frac{0.28}{0.020}=t[/tex]
⇒[tex]t=14[/tex]
Hence, the time period is 14 years.
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