SOLUTION: what interest rate is required for an investment with continuously compounded interest to double in 6 years

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Question 219555: what interest rate is required for an investment with continuously compounded interest to double in 6 years
Answer by jim_thompson5910(35256) About Me  (Show Source):
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A=Pe%5E%28rt%29 Start with the continuous compounding interest formula.


500=250%2Ae%5E%28r%2A6%29 Plug in A=500, P=250, and t=6.


500=250%2Ae%5E%286%2Ar%29 Rearrange the terms.


500%2F250=e%5E%286%2Ar%29 Divide both sides by 250.


2=e%5E%286%2Ar%29 Divide 500%2F250 to get 2.


ln%282%29=ln%28e%5E%286%2Ar%29%29 Take the natural log of both sides.


ln%282%29=6%2Ar%2Aln%28e%29 Pull down the exponent using the identity ln%28x%5Ey%29=y%2Aln%28x%29.


ln%282%29=6%2Ar%2A1 Evaluate ln%28e%29 to get 1.


ln%282%29=6%2Ar Multiply.


0.693147=6%2Ar Evaluate ln%282%29 to get 0.693147


0.693147%2F6=r Divide both sides by 6 to isolate 'r'


0.115525=r Divide 0.693147%2F6 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.