Question

suppose you were to deposit $224.00 into a savings account that earns 2.45% interest compounded continuously. use the continuously compounding interest formula $a = pe^{rt}$ to answer the following. a) how much money will be in the account after 10 years? round to the nearest cent. b) how much money will be in the account after 16 years? round your answer to the nearest cent.

Explanation:

Step1: Identify given values

We have \( P = 224 \), \( r = 2.45\% = 0.0245 \), and we'll use the formula \( A = Pe^{rt} \).

Part (a)

Step2: Substitute \( t = 10 \) into the formula

Substitute \( P = 224 \), \( r = 0.0245 \), \( t = 10 \) into \( A = Pe^{rt} \):
\( A = 224 \times e^{0.0245 \times 10} \)
First, calculate the exponent: \( 0.0245 \times 10 = 0.245 \)
Then, \( e^{0.245} \approx 1.2776 \) (using a calculator for \( e^{0.245} \))
Now, multiply by \( 224 \): \( 224 \times 1.2776 \approx 286.18 \)

Part (b)

Step3: Substitute \( t = 16 \) into the formula

Substitute \( P = 224 \), \( r = 0.0245 \), \( t = 16 \) into \( A = Pe^{rt} \):
\( A = 224 \times e^{0.0245 \times 16} \)
Calculate the exponent: \( 0.0245 \times 16 = 0.392 \)
Then, \( e^{0.392} \approx 1.479 \) (using a calculator for \( e^{0.392} \))
Now, multiply by \( 224 \): \( 224 \times 1.479 \approx 331.296 \approx 331.30 \)

Answer:

a) \(\$286.18\)
b) \(\$331.30\)