The function f(x) = 900 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 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) = 900 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 t= 10. (A) The present value is $ (Do not round until the final answer. Then round to the nearest cent as needed.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Continuous Compounding and Present Value Calculation
#### Problem Statement
The function \( f(x) = 900 \) represents the rate of flow of money in dollars per year. Assume a 10-year period at 4% compounded continuously. Find:
1. The present value (PV).
2. The accumulated amount of money flow at \( t = 10 \).
---
**(A)** *The Present Value Calculation:*
To find the present value (PV) of the continuous money flow, we use the formula for the present value of a continuous income stream:
\[ PV = \int_0^T f(t) e^{-rt} \, dt \]
Where:
- \( f(t) = 900 \)
- \( T = 10 \) years
- \( r = 0.04 \) (4% continuous compounding)
So, the present value is:
\[ PV = \int_0^{10} 900 e^{-0.04t} \, dt \]
*Do not round until the final answer. Then, round to the nearest cent as needed.*
---
To solve this integral:
\[ PV = 900 \int_0^{10} e^{-0.04t} \, dt \]
First, compute the integral:
\[ \int e^{-0.04t} \, dt \]
Using the antiderivative formula for \( e^{-kt} \):
\[ \int e^{-0.04t} \, dt = \frac{-1}{0.04} e^{-0.04t} = -25 e^{-0.04t} \]
Next, evaluate the definite integral from 0 to 10:
\[ PV = 900 \left[ -25 e^{-0.04t} \right]_0^{10} \]
\[ PV = 900 \left[ -25 e^{-0.4} - (-25) \right] \]
\[ PV = 900 \left[ -25 e^{-0.4} + 25 \right] \]
\[ PV = 900 \times 25 \left[ 1 - e^{-0.4} \right] \]
Given \( e^{-0.4} \approx 0.67032 \):
\[ PV = 900 \times 25 \left[ 1 - 0.67032 \right] \]
\[ PV = 900 \times 25 \times](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F756689fc-aa25-4045-a342-f553545a40f2%2F69e90726-3b7d-4979-8034-d3cfd0e5ee8c%2Fhsvscf_processed.png&w=3840&q=75)
Transcribed Image Text:### Continuous Compounding and Present Value Calculation
#### Problem Statement
The function \( f(x) = 900 \) represents the rate of flow of money in dollars per year. Assume a 10-year period at 4% compounded continuously. Find:
1. The present value (PV).
2. The accumulated amount of money flow at \( t = 10 \).
---
**(A)** *The Present Value Calculation:*
To find the present value (PV) of the continuous money flow, we use the formula for the present value of a continuous income stream:
\[ PV = \int_0^T f(t) e^{-rt} \, dt \]
Where:
- \( f(t) = 900 \)
- \( T = 10 \) years
- \( r = 0.04 \) (4% continuous compounding)
So, the present value is:
\[ PV = \int_0^{10} 900 e^{-0.04t} \, dt \]
*Do not round until the final answer. Then, round to the nearest cent as needed.*
---
To solve this integral:
\[ PV = 900 \int_0^{10} e^{-0.04t} \, dt \]
First, compute the integral:
\[ \int e^{-0.04t} \, dt \]
Using the antiderivative formula for \( e^{-kt} \):
\[ \int e^{-0.04t} \, dt = \frac{-1}{0.04} e^{-0.04t} = -25 e^{-0.04t} \]
Next, evaluate the definite integral from 0 to 10:
\[ PV = 900 \left[ -25 e^{-0.04t} \right]_0^{10} \]
\[ PV = 900 \left[ -25 e^{-0.4} - (-25) \right] \]
\[ PV = 900 \left[ -25 e^{-0.4} + 25 \right] \]
\[ PV = 900 \times 25 \left[ 1 - e^{-0.4} \right] \]
Given \( e^{-0.4} \approx 0.67032 \):
\[ PV = 900 \times 25 \left[ 1 - 0.67032 \right] \]
\[ PV = 900 \times 25 \times
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