A person places $8230 in an investment account earning an annual rate of 4.1%, compounded continuously. Using the formula V=Pe^(rt), where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 15 years. Answer:

Question

A person places  $8230 in an investment account earning an annual rate of  4.1%, compounded continuously. Using the formula  V=Pe^(rt), where  V is the value of the account in  t years,  P is the principal initially invested,  e is the base of a natural logarithm, and  r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 15 years. Answer:

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Identify Given Values: Identify the given values from the problem. Principal (P) = $8230 Rate (r) = 4.1% or 0.041 (as a decimal) Time (t) = 15 years We will use the formula V = Pe^(rt) to find the value (V) of the account after 15 years.Convert Rate to Decimal: Convert the percentage rate to a decimal. To convert a percentage to a decimal, divide by 100. r = 4.1% = 4.1/100 = 0.041Substitute Values into Formula: Substitute the values into the formula. Using the formula V = Pe^(rt), we substitute P = $8230, r = 0.041, and t = 15. V = 8230 * e^(0.041 * 15)Calculate Exponent: Calculate the exponent part of the formula. rt = 0.041 * 15 = 0.615 Now we have V = 8230 * e^0.615Calculate e Value: Calculate the value of e raised to the power of 0.615. Using a calculator, we find e^0.615 ≈ 1.84986.Multiply Principal by e Value: Multiply the principal by the value of e raised to the power of rt. V = 8230 * 1.84986 ≈ 15217.14Round to Nearest Cent: Round the result to the nearest cent. The value of the investment account after 15 years, rounded to the nearest cent, is approximately $15217.14.

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