Question

mario invested $6,000 in an account that pays 5% annual interest compounded annually. using the formula $a = p(1 + r)^t$, what is the approximate value of the account after 2.5 years? $6,118 $6,075 $6,456 $6,778

Explanation:

Step1: Identify given values

We know that the principal amount \( P = \$6000 \), the annual interest rate \( r = 5\% = 0.05 \), and the time \( t = 2.5 \) years. The formula for compound interest is \( A = P(1 + r)^t \).

Step2: Substitute values into the formula

Substitute \( P = 6000 \), \( r = 0.05 \), and \( t = 2.5 \) into the formula:
\[
A = 6000(1 + 0.05)^{2.5}
\]

Step3: Calculate \( (1 + 0.05)^{2.5} \)

First, calculate \( 1 + 0.05 = 1.05 \). Then, we need to find \( 1.05^{2.5} \). We can rewrite \( 2.5 \) as \( \frac{5}{2} \), so \( 1.05^{2.5} = \sqrt{1.05^5} \). Calculate \( 1.05^5 \approx 1.27628 \), then take the square root: \( \sqrt{1.27628} \approx 1.13 \) (a more accurate calculation: \( 1.05^{2.5}=e^{2.5\ln(1.05)}\approx e^{2.5\times0.04879}\approx e^{0.121975}\approx 1.13 \)).

Step4: Calculate the amount \( A \)

Now, multiply by the principal: \( A = 6000\times1.13 = 6780 \). The closest option to this value is \( \$6778 \).

Answer:

\$6,778