What was the initial amount deposited in an account if after 20 years the balance is $61,500, with an interest rate of 4.5% compounded continuously?

Use the formula to solve.

Question

What was the initial amount deposited in an account if after 20 years the balance is $61,500, with an interest rate of 4.5% compounded continuously?

Use the formula  to solve.

BlackTom AI · Accepted Answer

We start by identifying the formula for continuous compounding: A = P e^{rt}, where A is the amount after time t, P is the initial principal, r is the annual interest rate, and t is time in years.
First, note the given values: A = 61,500, r = 0.045 (4.5%), and t = 20 years. The goal is to solve for the initial amount P.
To isolate P, rearrange the formula: P = A e^{-rt}. This uses the fact that e^{rt} multiplied by e^{-rt} equals 1, so multiplying A by e^{-rt} undoes the accumulation.
Compute the exponent: rt = 0.045 × 20 = 0.9. Therefore, e^{-rt} = e^{-0.9}.
Next, approximate e^{-0.9}. Using a calculator or a known exponential table, e^{-0.9} ≈ 0.4060058.
Now multiply by A: P ≈ 61,500 × 0.4060058 ≈ 24,973. 
If you carry more precision, 61,500 × 0.4060058 ≈ 24,973.3, which rounds to about 24,973.3 dollars. 
In practice, you would report this as roughly $24,973, and you might see an approximation like about $25,000 depending on the rounding convention used. 
Keep in mind that the exact value depends on the precision of e^{-0.9} you employ, but all standard calculators yield a result near twenty-five thousand dollars for this scenario.

PRECALC141-033 2.8 Practice Problems

What was the initial amount deposited in an account if after 20 years the balance is $61,500, with an interest rate of 4.5% compounded continuously? Use the formula to solve.

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