find the interest rate for a $6000 deposit accumulating to $6966, compounded annually for 9 years.\nthe…

find the interest rate for a $6000 deposit accumulating to $6966, compounded annually for 9 years.\nthe interest rate is %.\n(do not round until the final answer. then round to two decimal places as needed.)

find the interest rate for a $6000 deposit accumulating to $6966, compounded annually for 9 years.\nthe interest rate is %.\n(do not round until the final answer. then round to two decimal places as needed.)

Answer

Explanation:

Step1: Recall compound interest formula

The formula for compound interest (annual compounding) is: $$A = P(1 + r)^t$$ Where:

  • $A = 6966$ (final amount),
  • $P = 6000$ (principal deposit),
  • $t = 9$ (number of years),
  • $r$ = annual interest rate (unknown)

Step2: Isolate the $(1+r)^t$ term

Divide both sides by $P$: $$\frac{A}{P} = (1 + r)^t$$ Substitute values: $$\frac{6966}{6000} = (1 + r)^9$$ Simplify left side: $$1.161 = (1 + r)^9$$

Step3: Solve for $1+r$

Take the 9th root of both sides (equivalent to raising to the $\frac{1}{9}$ power): $$1 + r = 1.161^{\frac{1}{9}}$$ Calculate the right-hand side: $$1 + r \approx 1.0165$$

Step4: Solve for $r$

Subtract 1 from both sides, then convert to percentage: $$r = 1.0165 - 1$$ $$r = 0.0165$$ $$r = 0.0165 \times 100 = 1.65%$$

Answer:

1.65