Question

ryan invested 5000 in an account that grows continuously at an annual rate of 2.5%. What will ryan’s investment be worth after 7 years? Round to the nearest cent

Answer 1

The formula for calculating the value of an investment that grows continuously is:

A = Pe^(rt)

Where:

A is the final amount

P is the principal amount

e is Euler's number (approximately 2.71828)

r is the annual interest rate (as a decimal)

t is the time in years

In this case, P = 5000, r = 0.025 (2.5% expressed as a decimal), and t = 7. Plugging these values into the formula, we get:

A = 5000 * e^(0.025*7) = 5000 * e^0.175 = 5000 * 1.19128 = 5956.40

Therefore, Ryan's investment will be worth $5,956.40 after 7 years. Rounded to the nearest cent, the answer is $5,956.40.

Answer 2

`~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^(rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$5000\\ r=rate\to 2.5\%\to (2.5)/(100)\dotfill &0.025\\ t=years\dotfill &7 \end{cases} \\\\\\ A = 5000e^(0.025\cdot 7) \implies A=5000e^(0.175) A \approx 5956.23`