**Population Growth Problem** Suppose a population is given by the equation: \[ P(t) = \frac{562}{1 + 75e^{-0.05t}} \] where \( P \) is in thousands and \( t \) is in years. When will this population reach 40,000? (Round to the nearest year) **Options:** - \( t = 28 \) - \( t = 15 \) - \( t = 41 \) - \( t = 35 \) **Solution Approach:** To solve for \( t \) when the population reaches 40,000: 1. Since \( P \) is in thousands, set \( P(t) = 40 \). 2. Solve the equation: \[ 40 = \frac{562}{1 + 75e^{-0.05t}} \] 3. Isolate \( e^{-0.05t} \) and solve for \( t \). This is a common question for learning exponential growth models, often used in biology, ecology, and demographics.
**Population Growth Problem** Suppose a population is given by the equation: \[ P(t) = \frac{562}{1 + 75e^{-0.05t}} \] where \( P \) is in thousands and \( t \) is in years. When will this population reach 40,000? (Round to the nearest year) **Options:** - \( t = 28 \) - \( t = 15 \) - \( t = 41 \) - \( t = 35 \) **Solution Approach:** To solve for \( t \) when the population reaches 40,000: 1. Since \( P \) is in thousands, set \( P(t) = 40 \). 2. Solve the equation: \[ 40 = \frac{562}{1 + 75e^{-0.05t}} \] 3. Isolate \( e^{-0.05t} \) and solve for \( t \). This is a common question for learning exponential growth models, often used in biology, ecology, and demographics.
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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Question
![**Population Growth Problem**
Suppose a population is given by the equation:
\[ P(t) = \frac{562}{1 + 75e^{-0.05t}} \]
where \( P \) is in thousands and \( t \) is in years. When will this population reach 40,000? (Round to the nearest year)
**Options:**
- \( t = 28 \)
- \( t = 15 \)
- \( t = 41 \)
- \( t = 35 \)
**Solution Approach:**
To solve for \( t \) when the population reaches 40,000:
1. Since \( P \) is in thousands, set \( P(t) = 40 \).
2. Solve the equation:
\[ 40 = \frac{562}{1 + 75e^{-0.05t}} \]
3. Isolate \( e^{-0.05t} \) and solve for \( t \).
This is a common question for learning exponential growth models, often used in biology, ecology, and demographics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca5e629a-9287-4ec7-9b80-8449992c5732%2Fd7e6f890-4ef2-4b36-8d0b-a5c421960b79%2Fl7gwztq_processed.png&w=3840&q=75)
Transcribed Image Text:**Population Growth Problem**
Suppose a population is given by the equation:
\[ P(t) = \frac{562}{1 + 75e^{-0.05t}} \]
where \( P \) is in thousands and \( t \) is in years. When will this population reach 40,000? (Round to the nearest year)
**Options:**
- \( t = 28 \)
- \( t = 15 \)
- \( t = 41 \)
- \( t = 35 \)
**Solution Approach:**
To solve for \( t \) when the population reaches 40,000:
1. Since \( P \) is in thousands, set \( P(t) = 40 \).
2. Solve the equation:
\[ 40 = \frac{562}{1 + 75e^{-0.05t}} \]
3. Isolate \( e^{-0.05t} \) and solve for \( t \).
This is a common question for learning exponential growth models, often used in biology, ecology, and demographics.
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