Question

An initial amount of $2500 is invested in an account at an interest rate of 6% per year, compounded continuously. Assuming that no withdrawals are made, find the amount in the account after five years. Do not round any intermediate computations, and round your answer to the nearest cent.

Answer 1

To find the amount in the account after five years with continuous compounding, we can use the formula for continuous compound interest:

A = P * e^(rt)

Where:

A = Amount in the account after time t

P = Initial principal (initial amount invested)

e = Euler's number (approximately 2.71828)

r = Interest rate per year (as a decimal)

t = Time in years

Given:

Initial amount (P) = $2500

Interest rate (r) = 6% per year (0.06 as a decimal)

Time (t) = 5 years

Plugging in the values into the formula:

A = 2500 * e^(0.06 * 5)

Calculating the exponent:

A = 2500 * e^(0.3)

Using a calculator or a computer, we can evaluate e^(0.3) to be approximately 1.34986.

Calculating the final amount:

A = 2500 * 1.34986

A ≈ $3374.65

Therefore, the amount in the account after five years, with continuous compounding, is approximately $3374.65

Step-by-step explanation:

Answer 2

To find the amount in the account after five years with continuous compounding, we can use the formula:

A = P * e^(rt)

Where:

A = the final amount including interest

P = the principal amount (initial deposit)

e = the mathematical constant approximately equal to 2.71828

r = the interest rate (in decimal form)

t = the number of years

In this case, the principal amount (P) is $2500, the interest rate (r) is 6% (or 0.06 in decimal form), and the number of years (t) is 5.

Plugging in these values into the formula, we get:

A = 2500 * e^(0.06 * 5)

Calculating this further:

A = 2500 * e^(0.3)

A ≈ 2500 * 1.34985880758

A ≈ $3374.65

Therefore, after five years of continuous compounding at a 6% interest rate, the amount in the account would be approximately $3374.65 to the nearest cent.