A bacteria culture initially contains 1000 bacteria and is growing at the rate of N' (t) = 2tt-2/3 bacteria per hour. Find the function N(t) that describes the number of bacteria in the culture after t hours and use it to determine how many bacteria there are after 27 hours. Type your answer in the space below. After 27 hours, there will be bacteria in the culture.
A bacteria culture initially contains 1000 bacteria and is growing at the rate of N' (t) = 2tt-2/3 bacteria per hour. Find the function N(t) that describes the number of bacteria in the culture after t hours and use it to determine how many bacteria there are after 27 hours. Type your answer in the space below. After 27 hours, there will be bacteria in the culture.
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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![### Understanding Bacterial Growth
A bacteria culture initially contains 1000 bacteria and is growing at the rate of
\[ N'(t) = 2t - t^{-\frac{2}{3}} \]
bacteria per hour. Find the function \( N(t) \) that describes the number of bacteria in the culture after \( t \) hours and use it to determine how many bacteria there are after 27 hours. Type your answer in the space below.
**After 27 hours, there will be __________ bacteria in the culture.**
---
To solve this problem, you need to integrate the rate function to find \( N(t) \), the total number of bacteria at any time \( t \).
1. **Given Differential Equation:**
\[ N'(t) = 2t - t^{-\frac{2}{3}} \]
2. **Integrate \( N'(t) \) to find \( N(t) \):**
\[ N(t) = \int (2t - t^{-\frac{2}{3}}) \, dt \]
3. **Determine the constants using initial conditions.**
Finally, substitute \( t = 27 \) into \( N(t) \) to find the number of bacteria after 27 hours.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbc3f4f4-d793-4827-8e3d-2f51535d2e99%2Fa7935a5c-4867-4bc5-98a2-84fd91ff86f4%2Fxa5n7vf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Bacterial Growth
A bacteria culture initially contains 1000 bacteria and is growing at the rate of
\[ N'(t) = 2t - t^{-\frac{2}{3}} \]
bacteria per hour. Find the function \( N(t) \) that describes the number of bacteria in the culture after \( t \) hours and use it to determine how many bacteria there are after 27 hours. Type your answer in the space below.
**After 27 hours, there will be __________ bacteria in the culture.**
---
To solve this problem, you need to integrate the rate function to find \( N(t) \), the total number of bacteria at any time \( t \).
1. **Given Differential Equation:**
\[ N'(t) = 2t - t^{-\frac{2}{3}} \]
2. **Integrate \( N'(t) \) to find \( N(t) \):**
\[ N(t) = \int (2t - t^{-\frac{2}{3}}) \, dt \]
3. **Determine the constants using initial conditions.**
Finally, substitute \( t = 27 \) into \( N(t) \) to find the number of bacteria after 27 hours.
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