Problem 1: A biologist friend has spent the past 25 months tracking the populations of two different species of fish. Let A(t) and B(t) be the size of each fish population t months after her observations began. While looking through her research journal, you find the following information: A"(t) 10 15 20 25 A' (10) = A'(20) = 0 -5 jotted down along with the graph of A"(t) shown to the right. --10 (a) When was the rate of change of the fish A population increasing? When was it decreasing? (b) When was the size of the fish A population increasing? When was it decreasing? (c) Which is bigger, the average rate of change in the fish A population between month 0 and month 5 or the average rate of change in the population between month 5 and month 10? (d) One of the results of your friend's research is that the size of the two fish populations are related. In particular, she notes that B(t) = 0.5A(t) +t² – 10t – 100. Which of the following statements was true about the fish B population at time t = 10: (i) After ten months, the fish B population was growing and the growth was speeding up. (ii) After ten months, the fish B population was growing and the growth was slowing down. (iii) After ten months, the fish B population was declining and the decline was speeding up. (iv) After ten months, the fish B population was declining and the decline was slowing down.
Problem 1: A biologist friend has spent the past 25 months tracking the populations of two different species of fish. Let A(t) and B(t) be the size of each fish population t months after her observations began. While looking through her research journal, you find the following information: A"(t) 10 15 20 25 A' (10) = A'(20) = 0 -5 jotted down along with the graph of A"(t) shown to the right. --10 (a) When was the rate of change of the fish A population increasing? When was it decreasing? (b) When was the size of the fish A population increasing? When was it decreasing? (c) Which is bigger, the average rate of change in the fish A population between month 0 and month 5 or the average rate of change in the population between month 5 and month 10? (d) One of the results of your friend's research is that the size of the two fish populations are related. In particular, she notes that B(t) = 0.5A(t) +t² – 10t – 100. Which of the following statements was true about the fish B population at time t = 10: (i) After ten months, the fish B population was growing and the growth was speeding up. (ii) After ten months, the fish B population was growing and the growth was slowing down. (iii) After ten months, the fish B population was declining and the decline was speeding up. (iv) After ten months, the fish B population was declining and the decline was slowing down.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 1:**
A biologist friend has spent the past 25 months tracking the populations of two different species of fish. Let \( A(t) \) and \( B(t) \) be the size of each fish population \( t \) months after her observations began. While looking through her research journal, you find the following information:
\[ A'(10) = A'(20) = 0 \]
jotted down along with the graph of \( A''(t) \) shown to the right.
**Questions:**
(a) When was the rate of change of the fish A population increasing? When was it decreasing?
(b) When was the size of the fish A population increasing? When was it decreasing?
(c) Which is bigger, the average rate of change in the fish A population between month 0 and month 5 or the average rate of change in the population between month 5 and month 10?
(d) One of the results of your friend’s research is that the size of the two fish populations are related. In particular, she notes that:
\[ B(t) = 0.5A(t) + t^2 - 10t - 100. \]
Which of the following statements was true about the fish B population at time \( t = 10 \):
(i) After ten months, the fish B population was growing and the growth was speeding up.
(ii) After ten months, the fish B population was growing and the growth was slowing down.
(iii) After ten months, the fish B population was declining and the decline was speeding up.
(iv) After ten months, the fish B population was declining and the decline was slowing down.
**Graph Description:**
The graph provided is of \( A''(t) \), which represents the second derivative of the fish A population over 25 months. It shows the changes in the concavity of the function \( A(t) \), indicating acceleration or deceleration in the rate of change. The graph starts below zero, increases and becomes positive after about month 7, peaks around month 15, and then declines, remaining positive.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F694d0814-403f-4047-9e1a-981c475298ee%2F1f053dd5-4265-43f4-8a11-147ef17d7265%2F7190xer_processed.gif&w=3840&q=75)
Transcribed Image Text:**Problem 1:**
A biologist friend has spent the past 25 months tracking the populations of two different species of fish. Let \( A(t) \) and \( B(t) \) be the size of each fish population \( t \) months after her observations began. While looking through her research journal, you find the following information:
\[ A'(10) = A'(20) = 0 \]
jotted down along with the graph of \( A''(t) \) shown to the right.
**Questions:**
(a) When was the rate of change of the fish A population increasing? When was it decreasing?
(b) When was the size of the fish A population increasing? When was it decreasing?
(c) Which is bigger, the average rate of change in the fish A population between month 0 and month 5 or the average rate of change in the population between month 5 and month 10?
(d) One of the results of your friend’s research is that the size of the two fish populations are related. In particular, she notes that:
\[ B(t) = 0.5A(t) + t^2 - 10t - 100. \]
Which of the following statements was true about the fish B population at time \( t = 10 \):
(i) After ten months, the fish B population was growing and the growth was speeding up.
(ii) After ten months, the fish B population was growing and the growth was slowing down.
(iii) After ten months, the fish B population was declining and the decline was speeding up.
(iv) After ten months, the fish B population was declining and the decline was slowing down.
**Graph Description:**
The graph provided is of \( A''(t) \), which represents the second derivative of the fish A population over 25 months. It shows the changes in the concavity of the function \( A(t) \), indicating acceleration or deceleration in the rate of change. The graph starts below zero, increases and becomes positive after about month 7, peaks around month 15, and then declines, remaining positive.
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