\begin{tabular}{l} Evaluating and Solving Exonential Functions \\ \hline TabaSue invested 17,950 dollars in a savings account that paid \( 4.9 \% \) interest compounded annually. Write \\ the exponential equation \( \boldsymbol{A} \) that represents TabaSue's investment where \( \boldsymbol{\Omega} \) is the accrued value of her \\ savings and \( t \) is the time of the investment in years. \\ \( A= \) \\ Determine how much money TabaSue will have after 18 years. \\ Answer: After 18 years, TabaSue will have \\ to the nearest penny/cent.) \\ Determine how long it will take for TabaSue's investment to double. \\ Answer: TabaSue's investment will have doubled in value after \\ the nearest tenth.) \end{tabular}

To solve the problem, we will first write the exponential equation for TabaSue's investment, then calculate the amount she will have after 18 years, and finally determine how long it will take for her investment to double. ### Step 1: Write the Exponential Equation The formula for compound interest is given by: \[ A = P(1 + r)^t \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( t \) is the number of years the money is invested or borrowed. For TabaSue's investment: - \( P = 17,950 \) - \( r = 4.9\% = 0.049 \) Thus, the equation becomes: \[ A = 17950(1 + 0.049)^t \] or \[ A = 17950(1.049)^t \] ### Step 2: Calculate the Amount After 18 Years Now we will calculate how much TabaSue will have after 18 years: \[ A = 17950(1.049)^{18} \] Calculating \( (1.049)^{18} \): \[ (1.049)^{18} \approx 2.406619 \] Now substituting this back into the equation: \[ A \approx 17950 \times 2.406619 \approx 43263.45 \] So, after 18 years, TabaSue will have approximately **$43,263.45**. ### Step 3: Determine How Long It Will Take for the Investment to Double To find out how long it will take for TabaSue's investment to double, we set \( A = 2P \): \[ 2 \times 17950 = 35900 \] Now we set up the equation: \[ 35900 = 17950(1.049)^t \] Dividing both sides by 17950: \[ 2 = (1.049)^t \] To solve for \( t \), we take the logarithm of both sides: \[ \log(2) = t \cdot \log(1.049) \] Now, solving for \( t \): \[ t = \frac{\log(2)}{\log(1.049)} \] Calculating the logarithms: \[ \log(2) \approx 0.3010 \] \[ \log(1.049) \approx 0.0212 \] Now substituting these values: \[ t \approx \frac{0.3010}{0.0212} \approx 14.2 \] So, TabaSue's investment will have doubled in value after approximately **14.2 years**. ### Summary of Answers - The exponential equation is \( A = 17950(1.049)^t \). - After 18 years, TabaSue will have approximately **$43,263.45**. - TabaSue's investment will have doubled in value after approximately **14.2 years**.