Me number of mosquitoes in a field after a major rainfall is modeled by the function M defined by M(t) = -t³ + 12t² + 144t, where t is the number of days after the rainfall ended and 0 ≤ t ≤ 18. a.) Using correct units, interpret the meaning of M' (15) = -171 in the context of the problem. b.) Based on the model, what is the absolute maximum number of mosquitoes in the field over the time interval 0 ≤ t ≤ 18? Justify your answer. want to not enoason ovi.noithoito mion a sedlo do or doin to c.) For what values of t is the rate of change of the number of mosquitoes in the field increasing? Javistni oris no" bas 'g to toivaded odt sdirdeab or wolod side odini eshins gnizaim od ni liri (5) erowans Toy tot asceno svi sviter to eviten steibril 20 E-= 0>x>8- svitizo d.) For 0 ≤ t ≤ 18, the number of bats in the field is modeled by the differentiable function B, where B is a function of the number of mosquitoes in the field. Based on the models, write an expression for the rate of change of the number of bats in the field at time t = a.
Me number of mosquitoes in a field after a major rainfall is modeled by the function M defined by M(t) = -t³ + 12t² + 144t, where t is the number of days after the rainfall ended and 0 ≤ t ≤ 18. a.) Using correct units, interpret the meaning of M' (15) = -171 in the context of the problem. b.) Based on the model, what is the absolute maximum number of mosquitoes in the field over the time interval 0 ≤ t ≤ 18? Justify your answer. want to not enoason ovi.noithoito mion a sedlo do or doin to c.) For what values of t is the rate of change of the number of mosquitoes in the field increasing? Javistni oris no" bas 'g to toivaded odt sdirdeab or wolod side odini eshins gnizaim od ni liri (5) erowans Toy tot asceno svi sviter to eviten steibril 20 E-= 0>x>8- svitizo d.) For 0 ≤ t ≤ 18, the number of bats in the field is modeled by the differentiable function B, where B is a function of the number of mosquitoes in the field. Based on the models, write an expression for the rate of change of the number of bats in the field at time t = a.
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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The number of mosquitos in a field after a major rainfall is modeled by the function M defined by M(t)=-t^3 + 12t^2 + 144t, where t is the number of days after the rainfall ended and 0
![**Problem 2: Modeling Mosquito Population Dynamics**
The population of mosquitoes in a field after a major rainfall is modeled by the function \( M \) defined by:
\[ M(t) = -t^3 + 12t^2 + 144t \]
where \( t \) is the number of days after the rainfall ended and \( 0 \leq t \leq 18 \).
**a.) Interpretation of \( M'(15) = -171 \) in Context**
Using correct units, interpret the meaning of \( M'(15) = -171 \) in the context of the problem.
**Answer:** The value \( M'(15) = -171 \) indicates the rate of change of the number of mosquitoes in the field on the 15th day after the rainfall ended. Since the derivative is negative, it suggests that the mosquito population is decreasing. Specifically, there is a decrease of 171 mosquitoes per day on the 15th day after the rainfall.
**b.) Maximum Number of Mosquitoes**
Based on the model, what is the absolute maximum number of mosquitoes in the field over the time interval \( 0 \leq t \leq 18 \)? Justify your answer.
**Answer:** To find the absolute maximum number of mosquitoes, we need to determine the critical points of \( M(t) \) by finding \( t \) where \( M'(t) = 0 \) within the interval \( 0 \leq t \leq 18 \). Then, we evaluate \( M(t) \) at those critical points and at the endpoints \( t = 0 \) and \( t = 18 \).
\[ M(t) = -t^3 + 12t^2 + 144t \]
\[ M'(t) = -3t^2 + 24t + 144 \]
Setting \( M'(t) = 0 \):
\[ -3t^2 + 24t + 144 = 0 \]
Solving for \( t \):
\[ t = -4 \ \text{or} \ t = 12 \ \text{(only the positive time is relevant)} \]
Evaluate \( M(t) \) at \( t = 0 \), \( t = 12 \), and \( t = 18 \):
\[ M(0) = 0 \]
\[ M(12) = -12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feed7c7e2-3561-4faf-afdd-7cb15025d3e1%2F8f756f85-63db-4fc0-a628-b15ed17dd744%2Fz7yty9l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Modeling Mosquito Population Dynamics**
The population of mosquitoes in a field after a major rainfall is modeled by the function \( M \) defined by:
\[ M(t) = -t^3 + 12t^2 + 144t \]
where \( t \) is the number of days after the rainfall ended and \( 0 \leq t \leq 18 \).
**a.) Interpretation of \( M'(15) = -171 \) in Context**
Using correct units, interpret the meaning of \( M'(15) = -171 \) in the context of the problem.
**Answer:** The value \( M'(15) = -171 \) indicates the rate of change of the number of mosquitoes in the field on the 15th day after the rainfall ended. Since the derivative is negative, it suggests that the mosquito population is decreasing. Specifically, there is a decrease of 171 mosquitoes per day on the 15th day after the rainfall.
**b.) Maximum Number of Mosquitoes**
Based on the model, what is the absolute maximum number of mosquitoes in the field over the time interval \( 0 \leq t \leq 18 \)? Justify your answer.
**Answer:** To find the absolute maximum number of mosquitoes, we need to determine the critical points of \( M(t) \) by finding \( t \) where \( M'(t) = 0 \) within the interval \( 0 \leq t \leq 18 \). Then, we evaluate \( M(t) \) at those critical points and at the endpoints \( t = 0 \) and \( t = 18 \).
\[ M(t) = -t^3 + 12t^2 + 144t \]
\[ M'(t) = -3t^2 + 24t + 144 \]
Setting \( M'(t) = 0 \):
\[ -3t^2 + 24t + 144 = 0 \]
Solving for \( t \):
\[ t = -4 \ \text{or} \ t = 12 \ \text{(only the positive time is relevant)} \]
Evaluate \( M(t) \) at \( t = 0 \), \( t = 12 \), and \( t = 18 \):
\[ M(0) = 0 \]
\[ M(12) = -12
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