Give the answer to this question correct to two decimal places. The compound interest on R23 269,00 invested at \( 10,50 \% \) interest per year, compounded semi-annually, is R10 022,43. The number of years for which the money was invested, is Select one: a. 7,00 . b. 4,10 . c. 16,46 . d. 3,50 .

To find the number of years for which the money was invested, 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 \( t \) 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 number of years the money is invested for. Given: - \( P = 23,269.00 \) - \( r = 10.50\% = 0.105 \) - \( n = 2 \) (compounded semi-annually) - \( A = 23,269.00 + 10,022.43 = 33,291.43 \) We can substitute these values into the formula and solve for \( t \). \[ 33,291.43 = 23,269.00 \left(1 + \frac{0.105}{2}\right)^{2t} \] Now, we can solve for \( t \) using the given options. Solve the equation by following steps: - step0: Solve for \(t\): \(33291.43=23269\left(1+\frac{0.105}{2}\right)^{2t}\) - step1: Simplify: \(33291.43=23269\left(\frac{421}{400}\right)^{2t}\) - step2: Swap the sides: \(23269\left(\frac{421}{400}\right)^{2t}=33291.43\) - step3: Multiply by the reciprocal: \(23269\left(\frac{421}{400}\right)^{2t}\times \frac{1}{23269}=33291.43\times \frac{1}{23269}\) - step4: Multiply: \(\left(\frac{421}{400}\right)^{2t}=\frac{3329143}{2326900}\) - step5: Take the logarithm of both sides: \(\log_{\frac{421}{400}}{\left(\left(\frac{421}{400}\right)^{2t}\right)}=\log_{\frac{421}{400}}{\left(\frac{3329143}{2326900}\right)}\) - step6: Evaluate the logarithm: \(2t=\log_{\frac{421}{400}}{\left(\frac{3329143}{2326900}\right)}\) - step7: Divide both sides: \(\frac{2t}{2}=\frac{\log_{\frac{421}{400}}{\left(\frac{3329143}{2326900}\right)}}{2}\) - step8: Divide the numbers: \(t=\frac{\log_{\frac{421}{400}}{\left(\frac{3329143}{2326900}\right)}}{2}\) - step9: Simplify: \(t=\frac{\log_{\frac{421}{400}}{\left(3329143\right)}-\log_{\frac{421}{400}}{\left(2326900\right)}}{2}\) The number of years for which the money was invested is approximately 3.50 years. Therefore, the correct answer is option d. 3.50.