Find the accumulated present value of the following continuous income stream at rate $R(t)$, for the given time $T$ and interest rate $k$, compounded continuously. \[ R(t)=0.07 t+400, \quad T=10, \quad k=4 \% \] The accumulated present value is $\$$ (Do not round until the final answer. Then round to the nearest cent as needed.)

Step 1 ：We are given a continuous income stream with rate \(R(t) = 0.07t + 400\), over a time period \(T = 10\), and an interest rate \(k = 4\%\) compounded continuously. We are asked to find the accumulated present value of this income stream.

Step 2 ：The formula for the accumulated present value of a continuous income stream is given by \(PV = \int_0^T R(t) e^{-kt} dt\).

Step 3 ：Substituting the given values into the formula, we get \(PV = \int_0^{10} (0.07t + 400) e^{-0.04t} dt\).

Step 4 ：Solving this integral, we find that the present value \(PV\) is approximately 3299.49.

Step 5 ：This means that the present value of the continuous income stream, discounted at a continuous compounding rate of 4%, is approximately $3299.49.

Step 6 ：Final Answer: \(\boxed{3299.49}\)