We need to find the time (in years) it takes for an investment of $700 to earn $300 in interest, given an interest rate of 11.6% compounded continuously. We need to round the final answer to the nearest whole number.

Applied MathematicsCompound InterestExponential FunctionsFinancial MathematicsLogarithms
2025/3/16

1. Problem Description

We need to find the time (in years) it takes for an investment of 700toearn700 to earn 300 in interest, given an interest rate of 11.6% compounded continuously. We need to round the final answer to the nearest whole number.

2. Solution Steps

The formula for continuous compounding is:
A=PertA = Pe^{rt}
where:
AA is the final amount (principal + interest)
PP is the principal amount (initial investment)
rr is the annual interest rate (as a decimal)
tt is the time in years
In this problem:
P=700P = 700
Interest earned =300= 300
Therefore, A=P+Interest=700+300=1000A = P + \text{Interest} = 700 + 300 = 1000
r=11.6%=0.116r = 11.6\% = 0.116
We need to find tt.
Substitute the known values into the formula:
1000=700e0.116t1000 = 700e^{0.116t}
Divide both sides by 700:
1000700=e0.116t\frac{1000}{700} = e^{0.116t}
107=e0.116t\frac{10}{7} = e^{0.116t}
Take the natural logarithm (ln) of both sides:
ln(107)=ln(e0.116t)\ln(\frac{10}{7}) = \ln(e^{0.116t})
ln(107)=0.116t\ln(\frac{10}{7}) = 0.116t
Solve for tt:
t=ln(107)0.116t = \frac{\ln(\frac{10}{7})}{0.116}
t=ln(10/7)0.116t = \frac{\ln(10/7)}{0.116}
t0.35667494390.116t \approx \frac{0.3566749439}{0.116}
t3.074783999t \approx 3.074783999
Round the result to the nearest whole number:
t3t \approx 3

3. Final Answer

3

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