SOLUTION: Recall that the formula for compounding interest is A = P(1+ r)t where A is the total accumulated amount in an account with principal P compounded at a annual interest rate r once

Algebra ->  Algebra  -> Exponential-and-logarithmic-functions -> SOLUTION: Recall that the formula for compounding interest is A = P(1+ r)t where A is the total accumulated amount in an account with principal P compounded at a annual interest rate r once       Log On

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Question 213724: Recall that the formula for compounding interest is A = P(1+ r)t where A is the total accumulated amount in an account with principal P compounded at a annual interest rate r once per year for t years.
a. How many years will it take for a principal of
P =1000 to reach an amount of A = 5000 when compounded once a year at an annual interest rate of 6%?
b. If the interest rate is doubled to 12%, what will the number of years be for the
principal of P =1000 to reach an amount of A = 5000?
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
a)

A=P%281%2Br%29%5Et Start with the given equation.


5000=1000%281%2B0.06%29%5Et Plug in P=1000, A=5000, and r=0.06


5000=1000%281.06%29%5Et Add


5000%2F1000=%281.06%29%5Et Divide both sides by 1000.


5=%281.06%29%5Et Divide


log%2810%2C%285%29%29=log%2810%2C%281.06%5Et%29%29 Take the log of both sides.


log%2810%2C%285%29%29=t%2Alog%2810%2C%281.06%29%29 Pull down the exponent.


0.69897=t%2Alog%2810%2C%281.06%29%29 Evaluate the left side


0.69897=t%2A0.02531 Evaluate the log on the right side


0.69897%2F0.02531=t Divide both sides by 0.02531.


t=27.61636 Divide and rearrange the equation.


So it will take about 27.61 years (a little over 27 and a half years) for $1,000 to become $5,000 at an interest rate of 6%

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b)

A=P%281%2Br%29%5Et Start with the given equation.


5000=1000%281%2B0.12%29%5Et Plug in P=1000, A=5000, and r=0.12


5000=1000%281.12%29%5Et Add


5000%2F1000=%281.12%29%5Et Divide both sides by 1000.


5=%281.12%29%5Et Divide


log%2810%2C%285%29%29=log%2810%2C%281.12%5Et%29%29 Take the log of both sides.


log%2810%2C%285%29%29=t%2Alog%2810%2C%281.12%29%29 Pull down the exponent.


0.69897=t%2Alog%2810%2C%281.12%29%29 Evaluate the left side


0.69897=t%2A0.04922 Evaluate the log on the right side


0.69897%2F0.04922=t Divide both sides by 0.04922.


t=14.20093 Divide and rearrange the equation.


So it will take about 14.2 years (a little over 14 years) for $1,000 to become $5,000 at an interest rate of 12%