A person's lung capacity can be modeled by the function C(t) = 250sin(t) + 2450, where C(t) represents the volume in mL present in the lungs aftert seconds. State the maximum value of this function over one full cycle, and explain what this value represents. Type your response here or upload a picture here
A person's lung capacity can be modeled by the function C(t) = 250sin(t) + 2450, where C(t) represents the volume in mL present in the lungs aftert seconds. State the maximum value of this function over one full cycle, and explain what this value represents. Type your response here or upload a picture here
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,...
Related questions
Question
![**Transcription of Image for Educational Website:**
---
### Question 11
A person's lung capacity can be modeled by the function \( C(t) = 250 \sin \left( \frac{2 \pi t}{5} \right) + 2450 \), where \( C(t) \) represents the volume in mL present in the lungs after \( t \) seconds. State the maximum value of this function over one full cycle, and explain what this value represents.
**Input Section:** [Type your response here or upload a picture here]
---
**Explanation:**
In this mathematical model, the function \( C(t) \) describes how the lung capacity or the volume of air in the lungs varies over time, measured in milliseconds (mL) and seconds (t). The function is specifically a sinusoidal function, which often represents periodic phenomena like breathing.
**Function Breakdown:**
- **Base Function:** \( C(t) = 250 \sin \left( \frac{2 \pi t}{5} \right) \)
- This part of the function has an amplitude of 250 mL, frequency coefficient of \( \frac{2 \pi}{5} \) indicating the cyclic nature of breathing.
- **Vertical Shift:** \( + 2450 \)
- This means the entire sinusoidal wave is shifted up by 2450 mL.
**Maximum Value:**
The sine function, \( \sin(\theta) \), oscillates between -1 and 1. Thus, the maximum value of \( \sin \left( \frac{2 \pi t}{5} \right) \) is 1.
- When \( \sin \left( \frac{2 \pi t}{5} \right) = 1 \):
\[
C_{\max}(t) = 250 \times 1 + 2450 = 250 + 2450 = 2700 \text{ mL}
\]
**Interpretation:**
The maximum value of 2700 mL represents the highest volume of air that the lungs can hold at any point during a breathing cycle. This suggests the peak or maximum lung capacity in this model is 2700 mL.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67663843-da70-4a78-a46e-b6153fa95861%2Feeeef8f0-cf01-4617-95e6-ccb0ef509f14%2F7tsr7i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription of Image for Educational Website:**
---
### Question 11
A person's lung capacity can be modeled by the function \( C(t) = 250 \sin \left( \frac{2 \pi t}{5} \right) + 2450 \), where \( C(t) \) represents the volume in mL present in the lungs after \( t \) seconds. State the maximum value of this function over one full cycle, and explain what this value represents.
**Input Section:** [Type your response here or upload a picture here]
---
**Explanation:**
In this mathematical model, the function \( C(t) \) describes how the lung capacity or the volume of air in the lungs varies over time, measured in milliseconds (mL) and seconds (t). The function is specifically a sinusoidal function, which often represents periodic phenomena like breathing.
**Function Breakdown:**
- **Base Function:** \( C(t) = 250 \sin \left( \frac{2 \pi t}{5} \right) \)
- This part of the function has an amplitude of 250 mL, frequency coefficient of \( \frac{2 \pi}{5} \) indicating the cyclic nature of breathing.
- **Vertical Shift:** \( + 2450 \)
- This means the entire sinusoidal wave is shifted up by 2450 mL.
**Maximum Value:**
The sine function, \( \sin(\theta) \), oscillates between -1 and 1. Thus, the maximum value of \( \sin \left( \frac{2 \pi t}{5} \right) \) is 1.
- When \( \sin \left( \frac{2 \pi t}{5} \right) = 1 \):
\[
C_{\max}(t) = 250 \times 1 + 2450 = 250 + 2450 = 2700 \text{ mL}
\]
**Interpretation:**
The maximum value of 2700 mL represents the highest volume of air that the lungs can hold at any point during a breathing cycle. This suggests the peak or maximum lung capacity in this model is 2700 mL.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education