When a child is born, her grandfather decides to put \( \$ 100 \) in an account that earns interest. He plans to make no other deposits or withdrawals for 18 years. When the child turns 18 years old, the money in the account will be a birthday gift. The grandfather is choosing between two options: Option 1: An account that grows by \( 10.5 \% \) each year. Option 2 : An account that grows by \( \$ 20 \) each year. a) Using Option 1, how much will be in the account after 18 years? b) Using Option 2 , how much will be in the account after 18 years? Which option will result in a better 18 th birthday gift? Explain your c) Ueasoning.

To solve this problem, we will calculate the amount in the account after 18 years for both options and then compare the results. ### a) Option 1: Account that grows by \( 10.5\% \) each year 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 time the money is invested for in years. For Option 1: - \( P = 100 \) - \( r = 0.105 \) - \( t = 18 \) Now, we can calculate the amount in the account after 18 years. ### b) Option 2: Account that grows by \( \$ 20 \) each year For Option 2, the amount in the account after 18 years can be calculated using a simple linear growth formula: \[ A = P + (d \times t) \] Where: - \( A \) is the total amount after \( t \) years. - \( P \) is the principal amount. - \( d \) is the amount added each year. - \( t \) is the number of years. For Option 2: - \( P = 100 \) - \( d = 20 \) - \( t = 18 \) Now, we can calculate the amount in the account after 18 years for both options. Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(100+\left(20\times 18\right)\) - step1: Multiply the numbers: \(100+360\) - step2: Add the numbers: \(460\) Calculate or simplify the expression \( 100*(1 + 0.105)^18 \). Calculate the value by following steps: - step0: Calculate: \(100\left(1+0.105\right)^{18}\) - step1: Add the numbers: \(100\times 1.105^{18}\) - step2: Convert the expressions: \(100\left(\frac{221}{200}\right)^{18}\) - step3: Simplify: \(100\times \frac{221^{18}}{200^{18}}\) - step4: Rewrite the expression: \(100\times \frac{221^{18}}{100^{18}\times 2^{18}}\) - step5: Reduce the numbers: \(1\times \frac{221^{18}}{100^{17}\times 2^{18}}\) - step6: Multiply the fractions: \(\frac{221^{18}}{100^{17}\times 2^{18}}\) ### a) Option 1: Amount after 18 years Using the compound interest formula, we calculated: \[ A = 100(1 + 0.105)^{18} \approx 603.28 \] So, after 18 years, the amount in the account for Option 1 will be approximately **\$603.28**. ### b) Option 2: Amount after 18 years Using the linear growth formula, we calculated: \[ A = 100 + (20 \times 18) = 100 + 360 = 460 \] So, after 18 years, the amount in the account for Option 2 will be **\$460**. ### c) Comparison and Reasoning Now, let's compare the two options: - **Option 1**: \$603.28 - **Option 2**: \$460 **Conclusion**: Option 1 results in a better 18th birthday gift, as it provides approximately \$603.28 compared to \$460 from Option 2. The higher growth rate of 10.5% compounded annually significantly increases the total amount over 18 years compared to the fixed annual addition of \$20 in Option 2.