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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![**Problem Statement:**
Rewrite the function \( P = 186(0.8)^t \) in the form \( P = P_0 e^{kt} \). (Round your answers to four decimal places.)
**Solution:**
1. **Determine \( P_0 \):**
Observe that \( P_0 \), the initial population or quantity, is the coefficient in front of the exponential function. In this equation, \( P_0 = 186 \).
2. **Find \( k \):**
The original equation uses a base of 0.8 for the exponential. To convert it into the base \( e \), the continuous growth rate \( k \) is found using the relationship:
\[
0.8 = e^k
\]
Taking the natural logarithm on both sides, we get:
\[
\ln(0.8) = k
\]
3. **Calculate \( k \):**
Compute the natural logarithm of 0.8:
\[
k \approx \ln(0.8) \approx -0.2231
\]
**Final Answers:**
\[
P_0 = 186
\]
\[
k \approx -0.2231
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d404c5d-4d55-46c0-aff4-29578c5eba85%2Ff332635e-5375-494b-b142-9531b6006dd4%2F9cvf10r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Rewrite the function \( P = 186(0.8)^t \) in the form \( P = P_0 e^{kt} \). (Round your answers to four decimal places.)
**Solution:**
1. **Determine \( P_0 \):**
Observe that \( P_0 \), the initial population or quantity, is the coefficient in front of the exponential function. In this equation, \( P_0 = 186 \).
2. **Find \( k \):**
The original equation uses a base of 0.8 for the exponential. To convert it into the base \( e \), the continuous growth rate \( k \) is found using the relationship:
\[
0.8 = e^k
\]
Taking the natural logarithm on both sides, we get:
\[
\ln(0.8) = k
\]
3. **Calculate \( k \):**
Compute the natural logarithm of 0.8:
\[
k \approx \ln(0.8) \approx -0.2231
\]
**Final Answers:**
\[
P_0 = 186
\]
\[
k \approx -0.2231
\]
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