Find the accumulated future value of the continuous income stream at rate $R(t)$, for the given time $T$, and interest rate $k$, compounded continuously.
\[
R(t)=\$ 40,000, T=16 \text { years, } k=4 \%
\]
Answer
Final Answer: The accumulated future value of the continuous income stream is approximately \$896,480.88. Therefore, the final answer is \(\boxed{896480.88}\).
Step 1 ：We are given a continuous income stream at rate \(R(t) = \$ 40,000\), for a time period of \(T = 16\) years, and an interest rate of \(k = 4\%\) compounded continuously. We are asked to find the accumulated future value of this income stream.
Step 2 ：The formula for the future value of a continuous income stream is given by: \[FV = \int_0^T R(t) e^{k(T-t)} dt\]
Step 3 ：In this case, the rate of income \(R(t)\) is constant and equal to \$40,000. The time \(T\) is 16 years and the interest rate \(k\) is 4%. Therefore, we can simplify the integral to: \[FV = \int_0^{16} 40000 e^{0.04(16-t)} dt\]
Step 4 ：Solving this integral, we find that the future value of the continuous income stream is approximately \$896,480.88.
Step 5 ：Final Answer: The accumulated future value of the continuous income stream is approximately \$896,480.88. Therefore, the final answer is \(\boxed{896480.88}\).
