Question 219555


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



{{{500=250*e^(r*6)}}} Plug in {{{A=500}}}, {{{P=250}}}, and {{{t=6}}}.



{{{500=250*e^(6*r)}}} Rearrange the terms.



{{{500/250=e^(6*r)}}} Divide both sides by {{{250}}}.



{{{2=e^(6*r)}}} Divide {{{500/250}}} to get {{{2}}}.



{{{ln(2)=ln(e^(6*r))}}} Take the natural log of both sides.



{{{ln(2)=6*r*ln(e)}}} Pull down the exponent using the identity {{{ln(x^y)=y*ln(x)}}}.



{{{ln(2)=6*r*1}}} Evaluate {{{ln(e)}}} to get {{{1}}}.



{{{ln(2)=6*r}}} Multiply.



{{{0.693147=6*r}}} Evaluate {{{ln(2)}}} to get {{{0.693147}}}



{{{0.693147/6=r}}} Divide both sides by {{{6}}} to isolate 'r'



{{{0.115525=r}}} Divide {{{0.693147/6}}} to get {{{0.115525}}}



{{{r=0.115525}}} Rearrange the equation.



{{{r=0.1155}}} Round to the nearest ten-thousandth.



So an interest rate of about 11.55% (don't forget to multiply by 100 to convert the decimal to a percent) will make the initial investment of $250 grow to $500 in 6 years.