Say you have invested \( \$ 35000 \) at a yearly interest rate of \( 5 \% \) compounded weekly. How many years will it take for our investment to double in size? Years \( = \)

To find out how many years it will take for the investment to double in size, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after \( n \) years, including interest. - \( P \) is the principal amount (initial investment). - \( r \) is the annual interest rate (in decimal form). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years. Given: - \( P = \$35000 \) - \( r = 5\% = 0.05 \) (in decimal form) - \( n = 52 \) (compounded weekly) We want to find \( t \) when \( A = 2P \). Substitute the values into the formula: \[ 2P = P \left(1 + \frac{0.05}{52}\right)^{52t} \] Simplify the equation: \[ 2 = \left(1 + \frac{0.05}{52}\right)^{52t} \] Now, we can solve for \( t \) using logarithms. Solve the equation by following steps: - step0: Solve for \(t\): \(2=\left(1+\frac{0.05}{52}\right)^{52t}\) - step1: Simplify: \(2=\left(\frac{1041}{1040}\right)^{52t}\) - step2: Swap the sides: \(\left(\frac{1041}{1040}\right)^{52t}=2\) - step3: Take the logarithm of both sides: \(\log_{\frac{1041}{1040}}{\left(\left(\frac{1041}{1040}\right)^{52t}\right)}=\log_{\frac{1041}{1040}}{\left(2\right)}\) - step4: Evaluate the logarithm: \(52t=\log_{\frac{1041}{1040}}{\left(2\right)}\) - step5: Divide both sides: \(\frac{52t}{52}=\frac{\log_{\frac{1041}{1040}}{\left(2\right)}}{52}\) - step6: Divide the numbers: \(t=\frac{\log_{\frac{1041}{1040}}{\left(2\right)}}{52}\) The solution to the equation is \( t = \frac{\log_{\frac{1041}{1040}}{(2)}}{52} \). Now, we can calculate the value of \( t \) to find out how many years it will take for the investment to double in size. Calculate the value by following steps: - step0: Calculate: \(\frac{\log_{\frac{1041}{1040}}{\left(2\right)}}{52}\) It will take approximately 13.87 years for the investment to double in size.