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?

• \\$6,075
• \\$6,118
• \\$6,456
• \\$6,778

Explanation:

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<reasoning>

Identify the given values

We are given the following values from the problem statement:

• Principal amount \(P = \$6,000\)
• Annual interest rate \(r = 5\% = 0.05\)
• Time in years \(t = 2.5\)
• Compound interest formula: \[A = P(1 + r)^t\]

Substitute the values into the formula

Substitute the given values into the compound interest formula:
\[A = 6000 \times (1 + 0.05)^{2.5}\]
\[A = 6000 \times (1.05)^{2.5}\]

Calculate the exponential term

Calculate the value of \((1.05)^{2.5}\):
\[(1.05)^{2.5} = (1.05)^2 \times (1.05)^{0.5}\]
\[(1.05)^2 = 1.1025\]
\[(1.05)^{0.5} = \sqrt{1.05} \approx 1.024695\]
\[(1.05)^{2.5} \approx 1.1025 \times 1.024695 \approx 1.129724\]

Calculate the final account value

Multiply the principal by the calculated exponential factor:
\[A \approx 6000 \times 1.129724\]
\[A \approx 6778.34\]

Comparing this result with the given choices, the approximate value is \(\$6,778\).
</reasoning>

<answer>
<mcq-option>(A) $6,075</mcq-option>
<mcq-option>(B) $6,118</mcq-option>
<mcq-option>(C) $6,456</mcq-option>
<mcq-correct>(D) $6,778</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the given values

We are given the following values from the problem statement:

• Principal amount \(P = \$6,000\)
• Annual interest rate \(r = 5\% = 0.05\)
• Time in years \(t = 2.5\)
• Compound interest formula: \[A = P(1 + r)^t\]

Substitute the values into the formula

Substitute the given values into the compound interest formula:
\[A = 6000 \times (1 + 0.05)^{2.5}\]
\[A = 6000 \times (1.05)^{2.5}\]

Calculate the exponential term

Calculate the value of \((1.05)^{2.5}\):
\[(1.05)^{2.5} = (1.05)^2 \times (1.05)^{0.5}\]
\[(1.05)^2 = 1.1025\]
\[(1.05)^{0.5} = \sqrt{1.05} \approx 1.024695\]
\[(1.05)^{2.5} \approx 1.1025 \times 1.024695 \approx 1.129724\]

Calculate the final account value

Multiply the principal by the calculated exponential factor:
\[A \approx 6000 \times 1.129724\]
\[A \approx 6778.34\]

Comparing this result with the given choices, the approximate value is \(\$6,778\).
</reasoning>

<answer>
<mcq-option>(A) $6,075</mcq-option>
<mcq-option>(B) $6,118</mcq-option>
<mcq-option>(C) $6,456</mcq-option>
<mcq-correct>(D) $6,778</mcq-correct>
</answer>

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