$$ \$ 800 $$ were deposited into an account wit a $$ 9.5 \% $$ interest rate, compounded continuously. How many years was it in the bank if the current amount is $$ \$ 6400 $$ ? $$ t=[?] \text { years } $$

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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 $$ P = \$800 $$,
- Interest rate $$ r = 9.5\% = 0.095 $$,
- Current amount $$ A = \$6400 $$.

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

Substitute the given values into the formula:

$$ 6400 = 800 e^{0.095t} $$

Now, we can solve for $$ t $$ using the formula for continuous compounding of interest.
Solve the equation by following steps:
- step0: Solve for $$t$$:
  $$6400=800e^{0.095t}$$
- step1: Convert the expressions:
  $$6400=800e^{\frac{19}{200}t}$$
- step2: Swap the sides:
  $$800e^{\frac{19}{200}t}=6400$$
- step3: Divide both sides:
  $$\frac{800e^{\frac{19}{200}t}}{800}=\frac{6400}{800}$$
- step4: Divide the numbers:
  $$e^{\frac{19}{200}t}=8$$
- step5: Take the logarithm of both sides:
  $$\ln{\left(e^{\frac{19}{200}t}\right)}=\ln{\left(8\right)}$$
- step6: Evaluate the logarithm:
  $$\frac{19}{200}t=\ln{\left(8\right)}$$
- step7: Multiply by the reciprocal:
  $$\frac{19}{200}t\times \frac{200}{19}=\ln{\left(8\right)}\times \frac{200}{19}$$
- step8: Multiply:
  $$t=\frac{200\ln{\left(8\right)}}{19}$$
- step9: Simplify:
  $$t=\frac{600\ln{\left(2\right)}}{19}$$
The time $$ t $$ in years is given by the formula:

$$ t = \frac{600 \ln(2)}{19} $$

Substitute the value of $$ \ln(2) $$ to find the numerical value of $$ t $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{600\ln{\left(2\right)}}{19}$$
The time $$ t $$ in years is approximately 21.89 years.
The time $$ t $$ in years is approximately 21.89 years.

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

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