The function f(x) = 1300 represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously. Find (A) the present value, and (B) the accumulated amount of money flow at t= 10. (A) The present value is $. (Do not round until the final answer. Then round to the nearest cent as needed.)
The function f(x) = 1300 represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously. Find (A) the present value, and (B) the accumulated amount of money flow at t= 10. (A) The present value is $. (Do not round until the final answer. Then round to the nearest cent as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
Homework:HW 16.3
2 of 8
![### Understanding Continuous Compounding and Money Flow Rate
The function \( f(x) = 1300 \) represents the rate of flow of money in dollars per year. We will work under the assumption of a 10-year period with an interest rate of 8% compounded continuously. Our objectives are to find:
(A) The present value.
(B) The accumulated amount of money flow at \( t = 10 \).
**(A) The present value is $ [input box] **
(Do not round until the final answer. Then round to the nearest cent as needed.)
### Explanation
1. **Compounded Continuously:**
Understanding continuous compounding is essential for financial calculations, especially when interest is compounded without interruption over a period.
2. **Function Representation:**
The given function \( f(x) = 1300 \) is a constant function representing the rate of flow of money. This means that every year, $1300 is the inflow rate.
3. **Interest Rate and Period:**
- **Interest Rate (continuous):** 8% or 0.08 as a decimal
- **Time Period (t):** 10 years
### Formulae to Note
\[ PV = \int_0^{10} 1300 \cdot e^{-0.08t} dt \]
Where \( PV \) is the present value, and \( e \) denotes the base of the natural logarithm.
### Calculation Steps
1. Integrate the given flow rate function with the exponential decay factor accounting for continuous compounding.
2. Compute the integral using appropriate limits and substitute to find the exact present value.
Following these steps accurately will yield the present value which is then to be rounded to the nearest cent as necessary.
By placing emphasis on accurate intermediate steps without rounding prematurely, we will ensure precision in the results for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c1127c9-3767-40bf-bfb8-794124c4dac3%2F6490fb20-4e68-4746-a0e6-a21ff6727b8a%2Fwod0aio_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Continuous Compounding and Money Flow Rate
The function \( f(x) = 1300 \) represents the rate of flow of money in dollars per year. We will work under the assumption of a 10-year period with an interest rate of 8% compounded continuously. Our objectives are to find:
(A) The present value.
(B) The accumulated amount of money flow at \( t = 10 \).
**(A) The present value is $ [input box] **
(Do not round until the final answer. Then round to the nearest cent as needed.)
### Explanation
1. **Compounded Continuously:**
Understanding continuous compounding is essential for financial calculations, especially when interest is compounded without interruption over a period.
2. **Function Representation:**
The given function \( f(x) = 1300 \) is a constant function representing the rate of flow of money. This means that every year, $1300 is the inflow rate.
3. **Interest Rate and Period:**
- **Interest Rate (continuous):** 8% or 0.08 as a decimal
- **Time Period (t):** 10 years
### Formulae to Note
\[ PV = \int_0^{10} 1300 \cdot e^{-0.08t} dt \]
Where \( PV \) is the present value, and \( e \) denotes the base of the natural logarithm.
### Calculation Steps
1. Integrate the given flow rate function with the exponential decay factor accounting for continuous compounding.
2. Compute the integral using appropriate limits and substitute to find the exact present value.
Following these steps accurately will yield the present value which is then to be rounded to the nearest cent as necessary.
By placing emphasis on accurate intermediate steps without rounding prematurely, we will ensure precision in the results for educational purposes.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
