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 7: Bacterial Growth Rate Calculation**
The doubling time for a certain bacteria population is 3 hours. What is the growth rate? Round to the nearest tenth of a percent.
**Explanation:**
To solve this problem, we need to determine the growth rate of a bacteria population given its doubling time. The growth rate can be calculated using the formula for exponential growth based on doubling time:
\[ \text{Growth rate (r)} = \left(\frac{\ln(2)}{\text{Doubling time (T)}}\right) \times 100 \]
Where \( \ln(2) \approx 0.693 \).
Let's calculate the growth rate step by step:
1. **Doubling Time (T)**: Given as 3 hours.
2. **Natural Logarithm of 2**: \( \ln(2) \approx 0.693 \).
**Calculation:**
\[ \text{Growth rate (r)} = \left(\frac{0.693}{3}\right) \times 100 \]
\[ \text{Growth rate (r)} \approx 0.231 \times 100 \]
\[ \text{Growth rate (r)} \approx 23.1\% \]
**Conclusion:**
The growth rate of the bacteria population is approximately 23.1% per hour.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F042a0375-4506-4632-a5f0-d842c57b4108%2Fe5a05e12-8d17-4022-a0a3-025cad07926c%2F7c74w1n_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7: Bacterial Growth Rate Calculation**
The doubling time for a certain bacteria population is 3 hours. What is the growth rate? Round to the nearest tenth of a percent.
**Explanation:**
To solve this problem, we need to determine the growth rate of a bacteria population given its doubling time. The growth rate can be calculated using the formula for exponential growth based on doubling time:
\[ \text{Growth rate (r)} = \left(\frac{\ln(2)}{\text{Doubling time (T)}}\right) \times 100 \]
Where \( \ln(2) \approx 0.693 \).
Let's calculate the growth rate step by step:
1. **Doubling Time (T)**: Given as 3 hours.
2. **Natural Logarithm of 2**: \( \ln(2) \approx 0.693 \).
**Calculation:**
\[ \text{Growth rate (r)} = \left(\frac{0.693}{3}\right) \times 100 \]
\[ \text{Growth rate (r)} \approx 0.231 \times 100 \]
\[ \text{Growth rate (r)} \approx 23.1\% \]
**Conclusion:**
The growth rate of the bacteria population is approximately 23.1% per hour.
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