Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
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![## Problem Statement:
**Express the exponential function \( f(t) = 20 \cdot 3^t \) in the form \( f(t) = a \cdot e^{rt} \). Also, find the doubling time of \( f(t) \).**
### Explanation:
1. **Converting to the form \( f(t) = a \cdot e^{rt} \):**
- Given the function \( f(t) = 20 \cdot 3^t \), we aim to express it in the form \( f(t) = a \cdot e^{rt} \).
- Recall that \( e^{rt} \) can be related to any exponential base by using natural logarithms: \( 3^t = e^{(t \ln 3)} \).
- Therefore, \( 3^t = e^{(t \ln 3)} \).
- Substitute this into the given function: \( f(t) = 20 \cdot e^{(t \ln 3)} \).
Thus, we have \( a = 20 \) and \( r = \ln 3 \). Hence, the function in the desired form is:
\[
f(t) = 20 \cdot e^{(\ln 3)t}
\]
2. **Finding the doubling time:**
- Doubling time \( T_d \) is the time it takes for the function's value to double.
- Therefore, we need \( f(T_d) = 2 \cdot f(0) \).
- At \( t = 0 \), \( f(0) = 20 \).
- Solve \( f(t) = 40 \) to find \( T_d \):
\[
40 = 20 \cdot e^{(\ln 3)T_d}
\]
- Divide both sides by 20:
\[
2 = e^{(\ln 3)T_d}
\]
- Take the natural logarithm of both sides:
\[
\ln 2 = (\ln 3)T_d
\]
- Solve for \( T_d \):
\[
T_d = \frac{\ln 2}{\ln 3}
\]
- Using approximate values for the logarithms:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8aee7c7d-285e-49c9-ad3b-faae5963cfad%2Fdce3c0da-de08-4395-a596-1491c5635dfb%2Fwkqury_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement:
**Express the exponential function \( f(t) = 20 \cdot 3^t \) in the form \( f(t) = a \cdot e^{rt} \). Also, find the doubling time of \( f(t) \).**
### Explanation:
1. **Converting to the form \( f(t) = a \cdot e^{rt} \):**
- Given the function \( f(t) = 20 \cdot 3^t \), we aim to express it in the form \( f(t) = a \cdot e^{rt} \).
- Recall that \( e^{rt} \) can be related to any exponential base by using natural logarithms: \( 3^t = e^{(t \ln 3)} \).
- Therefore, \( 3^t = e^{(t \ln 3)} \).
- Substitute this into the given function: \( f(t) = 20 \cdot e^{(t \ln 3)} \).
Thus, we have \( a = 20 \) and \( r = \ln 3 \). Hence, the function in the desired form is:
\[
f(t) = 20 \cdot e^{(\ln 3)t}
\]
2. **Finding the doubling time:**
- Doubling time \( T_d \) is the time it takes for the function's value to double.
- Therefore, we need \( f(T_d) = 2 \cdot f(0) \).
- At \( t = 0 \), \( f(0) = 20 \).
- Solve \( f(t) = 40 \) to find \( T_d \):
\[
40 = 20 \cdot e^{(\ln 3)T_d}
\]
- Divide both sides by 20:
\[
2 = e^{(\ln 3)T_d}
\]
- Take the natural logarithm of both sides:
\[
\ln 2 = (\ln 3)T_d
\]
- Solve for \( T_d \):
\[
T_d = \frac{\ln 2}{\ln 3}
\]
- Using approximate values for the logarithms:
\[
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