https://r2.erweima.ai/mathbot/image/1718158701622msobd67y.jpeg

To solve this problem, we need to calculate the present value of an investment that generates continuous income at a rate of $ f(t) = 2300 + 150t $ dollars per year, given a continuous compounding interest rate of 5%.

1. **Formulate the Present Value Integral:**

   The present value $ PV $ of a continuous income stream $ f(t) $ discounted at a continuous interest rate $ r $ is given by:
   \[
   PV = \int_{0}^{\infty} f(t) e^{-rt} \, dt
   \]
   Here, $ f(t) = 2300 + 150t $ and $ r = 0.05 $.

2. **Substitute $ f(t) $ into the Integral:**
   \[
   PV = \int_{0}^{\infty} (2300 + 150t) e^{-0.05t} \, dt
   \]

3. **Split the Integral:**
   \[
   PV = \int_{0}^{\infty} 2300 e^{-0.05t} \, dt + \int_{0}^{\infty} 150t e^{-0.05t} \, dt
   \]

4. **Solve Each Integral Separately:**

   For the first integral:
   \[
   \int_{0}^{\infty} 2300 e^{-0.05t} \, dt = 2300 \int_{0}^{\infty} e^{-0.05t} \, dt
   \]
   \[
   \int_{0}^{\infty} e^{-0.05t} \, dt = \left[ \frac{e^{-0.05t}}{-0.05} \right]_{0}^{\infty} = \left[ 0 - \frac{1}{-0.05} \right] = \frac{1}{0.05} = 20
   \]
   Therefore:
   \[
   2300 \times 20 = 46000
   \]

   For the second integral:
   \[
   \int_{0}^{\infty} 150t e^{-0.05t} \, dt
   \]
   This is a standard integral of the form $ \int_{0}^{\infty} t e^{-at} \, dt = \frac{1}{a^2} $:
   \[
   \int_{0}^{\infty} 150t e^{-0.05t} \, dt = 150 \int_{0}^{\infty} t e^{-0.05t} \, dt = 150 \times \frac{1}{0.05^2} = 150 \times 400 = 60000
   \]

5. **Add the Two Results:**
   \[
   PV = 46000 + 60000 = 106000
   \]

Therefore, the present value of the investment is **$106,000**.

Do you want details or have any questions? Here are some questions you might ask:

1. How do you calculate the present value of continuous cash flows?
2. Can you explain the continuous compounding interest formula?
3. What is the significance of the exponential discount factor $ e^{-rt} $?
4. How do you handle integrals involving time-dependent cash flows?
5. Can you show another example with different functions or interest rates?

**Tip:** When solving problems involving continuous cash flows, breaking down the integral into manageable parts can simplify the calculation.

Learn how to calculate the present value of an investment generating continuous income with a continuous compounding interest rate. This problem involves calculus, specifically integration techniques, and exponential function properties. Suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation