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You invested $6,000\$6,000 in a savings account, and after 77 years, the balance grew to $9,123.45\$9,123.45. The interest is compounded annually. What is the annual interest rate? Use the formula A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}, where AA is the balance (final amount), PP is the principal (starting amount), rr is the interest rate expressed as a decimal, nn is the number of times per year that the interest is compounded, and tt is the time in years. Round your answer to the nearest tenth.

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Q. You invested $6,000\$6,000 in a savings account, and after 77 years, the balance grew to $9,123.45\$9,123.45. The interest is compounded annually. What is the annual interest rate? Use the formula A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}, where AA is the balance (final amount), PP is the principal (starting amount), rr is the interest rate expressed as a decimal, nn is the number of times per year that the interest is compounded, and tt is the time in years. Round your answer to the nearest tenth.
  1. Given Values: We have: A = 9,123.459,123.45, P = 6,0006,000, t = 77 years, n = 11 (compounded annually). Formula: A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}
  2. Substitute into Formula: Substitute the values into the formula: 9,123.45=6,000(1+r1)179,123.45 = 6,000(1 + \frac{r}{1})^{1 \cdot 7}
  3. Simplify Equation: Simplify the equation: 9,123.45=6,000(1+r)79,123.45 = 6,000(1 + r)^7
  4. Divide by 66,000000: Divide both sides by 6,0006,000: (9,123.45/6,000)=(1+r)7(9,123.45 / 6,000) = (1 + r)^7 1.520575=(1+r)71.520575 = (1 + r)^7
  5. Take 77th Root: Take the 77th root of both sides to solve for (1+r) (1 + r) : (1.520575)(1/7)=1+r (1.520575)^{(1/7)} = 1 + r 1.060=1+r 1.060 = 1 + r
  6. Subtract 11 for rr: Subtract 11 from both sides to solve for rr: 1.0601=r1.060 - 1 = r r=0.060r = 0.060
  7. Convert to Percentage: Convert rr to a percentage: r=0.060imes100r = 0.060 imes 100 r=6.0extextpercentr = 6.0 ext{ extpercent}

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