Final answer:
To find the balance on Alyssa's 16th birthday, calculate the future value of two $2,500 deposits with 5.4% interest compounded monthly - one invested for 16 years and the other for 9 years. Then add both amounts together to find the total balance.
Step-by-step explanation:
To determine the balance in Alyssa's account on her 16th birthday, we will calculate the future value of two separate deposits that are compounded monthly. The first deposit occurs on the day she's born, and the second on her 7th birthday.
Calculating the Future Value of the First Deposit:
The formula for the future value of a compound interest account is: A = P(1 + r/n)^(nt), where:
- P is the principal amount (the initial amount of money)
- 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
For the first deposit of $2,500, at an interest rate of 5.4% compounded monthly, we have:
- P = $2,500
- r = 5.4/100 = 0.054
- n = 12 (monthly compound)
- t = 16 years (from birth until 16th birthday)
Plugging these values into the formula:
A1 = 2500(1 + 0.054/12)^(12*16)
Now, let's calculate the future value of the second deposit made on her 7th birthday:
Calculating the Future Value of the Second Deposit:
Since the second deposit is made 7 years after the first one, it will be compounded for 9 years (16 - 7 years). Using the same formula:
A2 = 2500(1 + 0.054/12)^(12*9)
Finally, we add both future values to obtain the total balance on Alyssa's account on her 16th birthday:
Total Balance = A1 + A2
Calculate A1 and A2 using a calculator, and then find their sum to determine the total balance.