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 determine the amount in the account for each option, we can use the formulas for compound interest and simple interest. a) For Option 1, where the account grows by \( 10.5\% \) each year, we use the formula for compound interest: \[ A = P(1 + r)^t \] Where \( A \) is the amount after \( t \) years, \( P = 100 \) (initial deposit), \( r = 0.105 \) (interest rate), and \( t = 18 \). \[ A = 100(1 + 0.105)^{18} \approx 100(5.476) \approx 547.60 \] So, the amount will be approximately \( \$547.60 \) after 18 years. b) For Option 2, where the account grows by \( \$20 \) each year, the total amount after 18 years can be calculated simply by adding \( \$20 \) for each of the 18 years: \[ A = P + (20 \times t) = 100 + (20 \times 18) = 100 + 360 = 460 \] Thus, the amount in the account will be \( \$460 \) after 18 years. c) Comparing both options, Option 1 yields \( \$547.60 \) while Option 2 yields \( \$460 \). Clearly, Option 1 results in a better 18th birthday gift. The power of compound interest is what makes the first option shine brighter; the money grows exponentially over time, while the second option simply adds a constant amount each year. It’s a fantastic lesson in the importance of understanding how interest works!