$$ \$ 1000 $$ are deposited in an account with a $$ 8.5 \% $$ interest rate, compounded continuously. What is the balance after 5 years? Begin by entering P, or the principal (initial investment). $$ P=\$[?] $$

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To find the balance after 5 years with continuous compounding, we can use the formula for continuous compounding:

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

Where:
- $$ A $$ is the balance after $$ t $$ years,
- $$ P $$ is the principal (initial investment),
- $$ r $$ is the interest rate,
- $$ t $$ is the time in years,
- $$ e $$ is the base of the natural logarithm.

Given:
- Principal (initial investment) $$ P = \$1000 $$,
- Interest rate $$ r = 8.5\% = 0.085 $$,
- Time $$ t = 5 $$ years.

Substitute the values into the formula:

$$ A = 1000 \times e^{0.085 \times 5} $$

Now, we can calculate the balance after 5 years.
Calculate the value by following steps:
- step0: Calculate:
  $$1000e^{0.085\times 5}$$
- step1: Multiply the numbers:
  $$1000e^{0.425}$$
- step2: Convert the expressions:
  $$1000e^{\frac{17}{40}}$$
- step3: Rewrite the expression:
  $$1000\sqrt[40]{e^{17}}$$
The balance after 5 years with continuous compounding is approximately $1529.59.
The balance after 5 years is approximately \$1,529.59.

are deposited in an account with a
interest rate, compounded
continuously.
What is the balance after 5 years?
Begin by entering P, or the principal
(initial investment).

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are deposited in an account with a interest rate, compounded continuously. What is the balance after 5 years? Begin by entering P, or the principal (initial investment).