Determine the balance of an account after 30 years where and compounded continuously. Round your answer to the nearest cent.

Question

Determine the balance of an account after 30 years where  and  compounded continuously. Round your answer to the nearest cent.

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BlackTom AI · Accepted Answer

The continuous compounding formula A = P e^{rt} applies, where P is the principal, r the annual interest rate (as a decimal), t the time in years, and A the account balance.

1) Relevant knowledge points and concepts
- Continuous compounding uses A = P e^{rt}. 
- Convert percent rate to decimal (7% → 0.07). 
- Carefully compute the exponential e^{rt} and multiply by the principal.

2) Complete step-by-step reasoning and intermediate calculations
- Given (from the context that leads to the provided correct answer): P = $2,500, r = 7% = 0.07, t = 30 years.
- Compute the exponent: rt = 0.07 × 30 = 2.1.
- Compute e^{2.1}. Break into known pieces: e^{2.1} = e^{2} × e^{0.1}.
  - e^{2} ≈ 7.389056099
  - e^{0.1} ≈ 1.105170918
  - Multiply: e^{2.1} ≈ 7.389056099 × 1.105170918 ≈ 8.166169912
- Multiply by the principal: A = 2500 × 8.166169912 ≈ 20,415.42478.
- Round to the nearest cent: A ≈ $20,415.42.

3) Why the correct answer is correct
- The formula A = P e^{rt} exactly models continuous compounding. Plugging in P = 2500, r = 0.07, t = 30 yields the computed value. The arithmetic above shows the direct evaluation that produces $20,415.42.

4) Why the other multiple-choice options are incorrect
- $40,830.85 is roughly double the correct answer (2 × $20,415.42) and does not match the continuous-compounding calculation.
- $7,144.13 and $7,609.81 are far smaller than the value produced by e^{2.1} growth and would correspond to much smaller rates, shorter times, or calculation errors.
- $166,715.83 is much larger and would require a vastly larger principal or rate; it does not match P = $2,500 with r = 7% over 30 years.

Conclusion: The correct balance after 30 years, compounded continuously, is $20,415.42, which matches the provided answer.

Determine the balance of an account after 30 years where and compounded continuously. Round your answer to the nearest cent. 单项选择题

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Determine the balance of an account after 30 years where and compounded continuously. Round your answer to the nearest cent. ​ 单项选择题