Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Calculating Bacteria Growth in Refrigerated Food**
The following mathematical problem demonstrates the calculation of bacteria growth in a refrigerated food product, using two functions that describe the number of bacteria and the change in temperature over time.
1. **Function Descriptions:**
- The number of bacteria \( N(T) \) in a refrigerated food product is given by the quadratic equation:
\[
N(T) = 25T^2 - 163T + 77, \quad 7 < T < 37
\]
where \( T \) represents the temperature of the food.
- Once the food is removed from the refrigerator, its temperature is governed by the linear function:
\[
T(t) = 6t + 1.6
\]
where \( t \) is the time in hours.
2. **Tasks:**
- **Composite Function**: Substitute the temperature function \( T(t) \) into the bacteria function \( N(T) \) to find \( N(T(t)) \).
- **Time Calculation**: Determine the time \( t \) when the bacteria count \( N(T(t)) \) reaches 7158, to at least two decimal places of accuracy.
3. **Solution Steps**:
- **Step 1**: Write the composite function \( N(T(t)) \):
\[
N(T(t)) = 25(6t + 1.6)^2 - 163(6t + 1.6) + 77
\]
- **Step 2**: Simplify the equation.
- **Step 3**: Solve \( N(T(t)) = 7158 \) for \( t \) ensuring the final value is accurate to at least two decimal places.
By working through these steps, one can determine the relationship between temperature change and bacteria growth over time and subsequently find the specific time when the bacteria count reaches a specified level.
**Calculation Box:**
- **Composite Function \( N(T(t)) \)**:
\[
N(T(t)) = \quad \rule{15cm}{0.5pt}
\]
- **Time Needed**:
\[
\text{Time Needed} = \quad \rule{10cm}{0.5pt} \quad \text{hours}
\]
This exercise requires a clear understanding of composing functions and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa78a88d7-60f0-4b9e-a272-6ff852667c68%2Fa11c0e1c-3ddf-4127-83be-54af9d376efb%2F0tm36fh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculating Bacteria Growth in Refrigerated Food**
The following mathematical problem demonstrates the calculation of bacteria growth in a refrigerated food product, using two functions that describe the number of bacteria and the change in temperature over time.
1. **Function Descriptions:**
- The number of bacteria \( N(T) \) in a refrigerated food product is given by the quadratic equation:
\[
N(T) = 25T^2 - 163T + 77, \quad 7 < T < 37
\]
where \( T \) represents the temperature of the food.
- Once the food is removed from the refrigerator, its temperature is governed by the linear function:
\[
T(t) = 6t + 1.6
\]
where \( t \) is the time in hours.
2. **Tasks:**
- **Composite Function**: Substitute the temperature function \( T(t) \) into the bacteria function \( N(T) \) to find \( N(T(t)) \).
- **Time Calculation**: Determine the time \( t \) when the bacteria count \( N(T(t)) \) reaches 7158, to at least two decimal places of accuracy.
3. **Solution Steps**:
- **Step 1**: Write the composite function \( N(T(t)) \):
\[
N(T(t)) = 25(6t + 1.6)^2 - 163(6t + 1.6) + 77
\]
- **Step 2**: Simplify the equation.
- **Step 3**: Solve \( N(T(t)) = 7158 \) for \( t \) ensuring the final value is accurate to at least two decimal places.
By working through these steps, one can determine the relationship between temperature change and bacteria growth over time and subsequently find the specific time when the bacteria count reaches a specified level.
**Calculation Box:**
- **Composite Function \( N(T(t)) \)**:
\[
N(T(t)) = \quad \rule{15cm}{0.5pt}
\]
- **Time Needed**:
\[
\text{Time Needed} = \quad \rule{10cm}{0.5pt} \quad \text{hours}
\]
This exercise requires a clear understanding of composing functions and
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