Final answer:
Rick's father needs to deposit initially approximately $12,420.45 into a savings account that offers 8% interest compounded daily for 16 years to accumulate $49,000.
Step-by-step explanation:
To determine how much Rick's father should deposit initially into a savings account that offers 8 percent interest compounded daily to reach $49,000 in 16 years, we use the formula for compound interest:
A = P(1 + r/n)nt
Where:
- A is the future value of the investment/loan, including interest,
- P is the principal investment amount (the initial deposit),
- r is the annual interest rate (decimal),
- n is the number of times that interest is compounded per year, and
- t is the number of years the money is invested or borrowed.
In this case, A is $49,000, r is 0.08 (8 percent), n is 365 (since the interest is compounded daily), and t is 16 years.
Now solving for P:
49000 = P(1 + 0.08/365)365*16
Let's first calculate the compound factor:
(1 + 0.08/365)365*16
After calculating the compound factor, we can rearrange the formula to solve for P:
P = 49000 / (1 + 0.08/365)365*16
Calculating this will give us the amount Rick's father needs to deposit.
Using a calculator:
P = 49000 / 3.9463
P ≈ $12420.45
Therefore, Rick's father needs to deposit approximately $12,420.45 initially to reach a balance of $49,000 in 16 years, assuming the interest is compounded daily at a rate of 8%.