Find the time required for an investment of 5000 dollars to grow to 6000 dollars at an interest rate of 7.5 percent per year, compounded quarterly. Your answer is \( t= \) Question Help: Video Submit Question

To find the time required for an investment of $5000 to grow to $6000 at an interest rate of 7.5% per year, compounded quarterly, 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 time the money is invested for, in years. Given: - \( P = 5000 \) dollars - \( A = 6000 \) dollars - \( r = 7.5\% = 0.075 \) (in decimal form) - \( n = 4 \) (compounded quarterly) We can substitute these values into the formula and solve for \( t \). Let's calculate it. Solve the equation by following steps: - step0: Solve for \(t\): \(6000=5000\left(1+\frac{0.075}{4}\right)^{4t}\) - step1: Simplify: \(6000=5000\left(\frac{163}{160}\right)^{4t}\) - step2: Swap the sides: \(5000\left(\frac{163}{160}\right)^{4t}=6000\) - step3: Multiply by the reciprocal: \(5000\left(\frac{163}{160}\right)^{4t}\times \frac{1}{5000}=6000\times \frac{1}{5000}\) - step4: Multiply: \(\left(\frac{163}{160}\right)^{4t}=\frac{6}{5}\) - step5: Take the logarithm of both sides: \(\log_{\frac{163}{160}}{\left(\left(\frac{163}{160}\right)^{4t}\right)}=\log_{\frac{163}{160}}{\left(\frac{6}{5}\right)}\) - step6: Evaluate the logarithm: \(4t=\log_{\frac{163}{160}}{\left(\frac{6}{5}\right)}\) - step7: Divide both sides: \(\frac{4t}{4}=\frac{\log_{\frac{163}{160}}{\left(\frac{6}{5}\right)}}{4}\) - step8: Divide the numbers: \(t=\frac{\log_{\frac{163}{160}}{\left(\frac{6}{5}\right)}}{4}\) - step9: Simplify: \(t=\frac{\log_{\frac{163}{160}}{\left(6\right)}-\log_{\frac{163}{160}}{\left(5\right)}}{4}\) The time required for an investment of $5000 to grow to $6000 at an interest rate of 7.5% per year, compounded quarterly is given by: \[ t = \frac{\log_{\frac{163}{160}}{(6)} - \log_{\frac{163}{160}}{(5)}}{4} \] This expression represents the time in years. Let's calculate the exact value of \( t \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\log_{\frac{163}{160}}{\left(6\right)}-\log_{\frac{163}{160}}{\left(5\right)}\right)}{4}\) - step1: Remove the parentheses: \(\frac{\log_{\frac{163}{160}}{\left(6\right)}-\log_{\frac{163}{160}}{\left(5\right)}}{4}\) The time required for an investment of $5000 to grow to $6000 at an interest rate of 7.5% per year, compounded quarterly is approximately 2.453674 years.