Answer:
[tex]t\approx6\hspace{3}years[/tex]
Explanation:
When interest is compounded annually, we can use the following formula to calculate the amount in the account at the end of a given time period.:
[tex]FV=PV(1+r)^t[/tex]
Where:
[tex]FV=Future\hspace{3}value=9140.20\\PV=Present\hspace{3}value=5000\\r=Interest\hspace{3}rate=11.2\%=0.112\\t=Time[/tex]
Let's solve the previous equation for t:
Divide both sides by PV:
[tex]\frac{FV}{PV} =\frac{PV}{PV} (1+r)^t\\\\\frac{FV}{PV} = (1+r)^t[/tex]
Take the natural logarithm of both sides:
[tex]log(\frac{FV}{PV}) = log( (1+r)^t)\\\\Use\hspace{3}the\hspace{3}identity\hspace{5}log(a^b)=b*log(a)\\\\log(\frac{FV}{PV}) =t* log(1+r)\\\\Divide\hspace{3}both\hspace{3}sides\hspace{3}by\hspace{3}log(1+r)\\\\t=\frac{log(\frac{FV}{PV})}{log(1+r)}[/tex]
Replace the data provided by the problem:
[tex]t=\frac{log(\frac{9140.20}{5000})}{log(1+0.112)}[/tex]
[tex]t=5.682396777\approx 6\hspace{3}years[/tex]
