To determine the amount of annual deposits Derek needs to make into his retirement account starting on his 29th birthday and ending on his 65th birthday (for a total of 37 years of deposits), we can use the future value of an ordinary annuity formula.
The future value \( FV \) of an ordinary annuity can be calculated using the formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) is the future value of the annuity (in this case, $3,816,699.00).
- \( P \) is the annual deposit (what we are trying to find).
- \( r \) is the annual interest rate (10.00% or 0.10).
- \( n \) is the number of deposits (from age 29 to age 65 is 37 years).
Rearranging the formula to solve for \( P \):
\[ P = \frac{FV}{\frac{(1 + r)^n - 1}{r}} \]
Plugging in the values:
- \( FV = 3,816,699.00 \)
- \( r = 0.10 \)
- \( n = 37 \)
We first calculate \( (1 + r)^n \):
\[ (1 + 0.10)^{37} = 1.10^{37} \]
Calculating \( 1.10^{37} \):
\[ 1.10^{37} \approx 14.35 \]
Now, plug this value back into the formula:
\[ \frac{(1 + r)^n - 1}{r} = \frac{14.35 - 1}{0.10} = \frac{13.35}{0.10} = 133.5 \]
Now substitute this value into the equation for \( P \):
\[ P = \frac{3,816,699.00}{133.5} \approx 28,528.96 \]
Thus, the annual deposit Derek needs to make is approximately:
\[ \boxed{28,528.96} \]
So, Derek must make annual deposits of about $28,528.96 starting on his 29th birthday until he turns 65 to reach his retirement goal of $3,816,699.00.