The function $f(x)=0.08 x+100$ represents the rate of flow of money in dollars per year. Assume a 10-year period at $4 \%$ compounded continuously. Find (A) the present value, and $(B)$ the accumulated amount of money flow at $\mathrm{t}=10$.
Answer
Final Answer: The present value of the money flow is \(\boxed{827.28}\) and the accumulated amount of money flow at \(t=10\) is \(\boxed{1234.81}\).
Step 1 ：We are given a function that represents the rate of flow of money in dollars per year, \(f(x) = 0.08x + 100\), over a 10-year period with a 4% interest rate compounded continuously.
Step 2 ：We can calculate the present value of a continuous money flow using the formula: \(PV = \int_0^T f(t) e^{-rt} dt\), where \(PV\) is the present value, \(f(t)\) is the rate of money flow at time \(t\), \(r\) is the interest rate, and \(T\) is the time period.
Step 3 ：Substituting the given values into the formula, we get \(PV = \int_0^{10} (0.08t + 100) e^{-0.04t} dt\).
Step 4 ：Calculating the integral, we find that the present value of the money flow is approximately \$827.28.
Step 5 ：We can also calculate the accumulated amount of a continuous money flow using the formula: \(A = \int_0^T f(t) e^{rt} dt\), where \(A\) is the accumulated amount, \(f(t)\) is the rate of money flow at time \(t\), \(r\) is the interest rate, and \(T\) is the time period.
Step 6 ：Substituting the given values into the formula, we get \(A = \int_0^{10} (0.08t + 100) e^{0.04t} dt\).
Step 7 ：Calculating the integral, we find that the accumulated amount of money flow at \(t=10\) is approximately \$1234.81.
Step 8 ：Final Answer: The present value of the money flow is \(\boxed{827.28}\) and the accumulated amount of money flow at \(t=10\) is \(\boxed{1234.81}\).
