$$ \$ 600 $$ were deposited into an account with a $$ 5 \% $$ interest rate, compounded continuously. How many years was it in the bank if the current amount is $$ \$ 6000 $$ ? $$ t=[?] $$ years Round to the nearest year.

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To solve this problem, we can use the formula for continuous compounding of interest:

$$ A = P e^{rt} $$

where:
- $$ A $$ is the current amount in the account,
- $$ P $$ is the initial deposit,
- $$ r $$ is the interest rate,
- $$ t $$ is the time in years.

Given:
- Initial deposit $$ P = \$600 $$,
- Interest rate $$ r = 5\% = 0.05 $$,
- Current amount $$ A = \$6000 $$.

We need to find the time $$ t $$ in years.

Substitute the given values into the formula:

$$ 6000 = 600 e^{0.05t} $$

Now, we can solve for $$ t $$ using the formula for continuous compounding of interest.
Solve the equation by following steps:
- step0: Solve for $$t$$:
  $$6000=600e^{0.05t}$$
- step1: Convert the expressions:
  $$6000=600e^{\frac{1}{20}t}$$
- step2: Swap the sides:
  $$600e^{\frac{1}{20}t}=6000$$
- step3: Divide both sides:
  $$\frac{600e^{\frac{1}{20}t}}{600}=\frac{6000}{600}$$
- step4: Divide the numbers:
  $$e^{\frac{1}{20}t}=10$$
- step5: Take the logarithm of both sides:
  $$\ln{\left(e^{\frac{1}{20}t}\right)}=\ln{\left(10\right)}$$
- step6: Evaluate the logarithm:
  $$\frac{1}{20}t=\ln{\left(10\right)}$$
- step7: Multiply by the reciprocal:
  $$\frac{1}{20}t\times 20=\ln{\left(10\right)}\times 20$$
- step8: Multiply:
  $$t=20\ln{\left(10\right)}$$
The solution to the equation is $$ t = 20 \ln{(10)} $$.

Now, we can calculate the value of $$ t $$ by substituting the value of $$ \ln{(10)} $$ into the equation.

$$ t = 20 \ln{(10)} $$

$$ t \approx 20 \times 2.3026 $$

$$ t \approx 46.052 $$

Rounding to the nearest year, the time $$ t $$ is approximately 46 years.
It took approximately 46 years for the \$600 to grow to \$6000 with continuous compounding at a 5% interest rate.

were deposited into an account with
a interest rate, compounded
continuously. How many years was it in
the bank if the current amount is ?
years
Round to the nearest year.

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were deposited into an account with a interest rate, compounded continuously. How many years was it in the bank if the current amount is ? years Round to the nearest year.