Given the following information, solve for the unknown annual interest rate (\(r\)):An investment of \$5,000 grows to \$7,500 in 5 years with interest compounded annually. What is the annual interest rate?A) 8.45% B) 10.00% C) 12.47% D) 15.00%

• A. 15.00%
• B. 10.00%
• C. 12.47%
• D. 8.45%

Answer: (C)

Verified step by step guidance

1. Step 1: Identify the formula for compound interest. The formula is: \( A = P(1 + r)^t \), where \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal form), and \( t \) is the time in years.
2. Step 2: Substitute the given values into the formula. Here, \( A = 7500 \), \( P = 5000 \), and \( t = 5 \). The equation becomes: \( 7500 = 5000(1 + r)^5 \).
3. Step 3: Divide both sides of the equation by \( P \) (5000) to isolate \( (1 + r)^5 \). This simplifies to: \( \frac{7500}{5000} = (1 + r)^5 \), or \( 1.5 = (1 + r)^5 \).
4. Step 4: Take the fifth root (or raise both sides to the power of \( \frac{1}{5} \)) to solve for \( 1 + r \). This gives: \( (1.5)^{\frac{1}{5}} = 1 + r \).
5. Step 5: Subtract 1 from both sides to isolate \( r \). The equation becomes: \( r = (1.5)^{\frac{1}{5}} - 1 \). Convert \( r \) into a percentage by multiplying the result by 100.