Use the “rule of 72” to estimate the doubling time (in years) for the interest rate, and then calculate it exactly. (Round your answers to two decimal places.): 9% compounded annually.

The "rule of 72" is a shortcut method for estimating the doubling time of an investment, given a fixed interest rate. It works by dividing 72 by the interest rate to get an estimate of the number of years it would take for the investment to double in value. So for an interest rate of 9% compounded annually, we can use the rule of 72 and divide 72 by 9 to get an estimate of the doubling time: 72/9 = 8 years This suggests that it would take \approx imately 8 years for an investment with a 9% annual interest rate to double in value. To calculate the exact doubling time, we can use the formula for compound interest: A = P(1 + r/n)^(nt) where: - A is the final amount - P is the starting principal - r is the annual interest rate (as a decimal) - n is the number of times the interest is compounded per year - t is the number of years In this case, we are given an interest rate of 9% compounded annually (n = 1), and we want to find the time it takes for the starting principal to double (A/P = 2). Plugging these values into the formula, we get: 2P = P(1 + 0.09/1)^(1t) Simplifying, we can cancel out the P on both sides of the equation: 2 = (1.09)^t Taking the logarithm of both sides, we get: log(2) = log(1.09^t) Using the power rule of logarithms, we can bring the exponent t down: log(2) = t * log(1.09) Finally, we can solve for t by dividing both sides by log(1.09): t = log(2) / log(1.09) Using a calculator, we can evaluate this expression to get: t ? 7.97 Rounding to two decimal places, we can say that the exact doubling time for an interest rate of 9% compounded annually is \approx imately 7.97 years.