Question 219496


{{{A=Pe^(rt)}}} Start with the continuous compounding interest formula.



{{{2P=P*e^(0.08*t)}}} Plug in {{{A=2P}}} (since we want to double our investment), and {{{r=0.08}}} (note: 0.08 is the decimal equivalent of 8%).



{{{(2P)/P=e^(0.08*t)}}} Divide both sides by {{{800}}}.



{{{2=e^(0.08*t)}}} Reduce.



{{{ln(2)=ln(e^(0.08*t))}}} Take the natural log of both sides.



{{{ln(2)=0.08*t*ln(e)}}} Pull down the exponent using the identity {{{ln(x^y)=y*ln(x)}}}.



{{{ln(2)=0.08*t*1}}} Evaluate {{{ln(e)}}} to get {{{1}}}.



{{{ln(2)=0.08*t}}} Multiply.



{{{0.693147=0.08*t}}} Evaluate {{{ln(2)}}} to get {{{0.693147}}}



{{{0.693147/0.08=t}}} Divide both sides by {{{0.08}}} to isolate 't'



{{{8.664338=t}}} Divide {{{0.693147/0.08}}} to get {{{8.664338}}}



{{{t=8.664338}}} Rearrange the equation.



{{{t=8.66}}} Round to the nearest hundredth.



So it will take about 8.66 years for an initial investment to double if it is compounded continuously at an interest rate of 8%.