Question

If ​$7000 is invested at 3​% compounded​ continuously, what is the amount after 7 ​years?
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Part 1
The amount after 7 years will be ​$
  
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Answer 1

To calculate the amount of money after 7 years with continuous compounding, you can use the formula:

\[
A = Pe^{rt}
\]

where:
- \( A \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount (the initial amount of money),
- \( r \) is the annual interest rate (decimal),
- \( t \) is the time the money is invested for in years, and
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).

Given:
- \( P = 7000 \)
- \( r = 0.03 \) (3% as a decimal)
- \( t = 7 \)

Now, plug in the values into the formula:

\[
A = 7000 \times e^{0.03 \times 7}
\]

Calculating the exponent:

\[
0.03 \times 7 = 0.21
\]

Next, calculate \( e^{0.21} \):

\[
e^{0.21} \approx 1.2337
\]

Now, multiply this by the principal:

\[
A = 7000 \times 1.2337 \approx 8635.90
\]

Thus, the amount after 7 years will be approximately:

\[
\boxed{8635.90}
\]