Question 3 Each scenario below describes a situation which can be modeled by a differential equation. Do the following for each scenario: 1. Choose the correct equation. 2. Show and explain your reasoning for your choice. c. Suppose you are done making a soup. It was too hot and after checking the temperatureyou realized that it was about 100°C. You decide to let it cool in your dining room withtemperature 25°C. The Newton's law of Cooling states that the rate of cooling of an objectis proportional to the temperature difference between the object and its surroundings,provided that this difference is not too large. Let T(t) represents the temperature of abowl of soup set out in the dining room every minute.(i)(ii)(iii)dTdtdTdtdTdt=100k (T25)-= k(T - 25)= kT-100 a. A biologist is making a bacteria culture starting with 100 cells in a petri dish. Thebiologist is recording the number of bacteria every hour. Let N(t) be the number ofbacteria in the petri dish after t hours. After many recording the biologist finds that thechange in the number of bacteria is proportional to the square of the number of bacteriapresent at the time of recording.(i)dNdt€dNdtdNdtdsdtb. A youtuber decides to analyze her subscribers growth. Let S(t) be the number of sub-scribers after t months. Using an analysis software, she notices that the change in thenumber of subscribers is proportional to the square root of the age (in months) of heraccount. Note that when her account was 9 months old her subscribers had increased by300.(i)dSdt= k(100 – N)²dsdt= k(N² – 100)= kN²=100√t3000√t= k√(t+300)c. Suppose you are done making a soup. It was too hot and after checking the temperatureyou realized that it was about 100°C. You decide to let it cool in your dining room withtemperature 25°C. The Newton's law of Cooling states that the rate of cooling of an objectis proportional to the temperature difference between the object and its surroundings,provided that this difference is not too large. Let T(t) represents the temperature of abowl of soup set out in the dining room every minute.

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c. Suppose you are done making a soup. It was too hot and after checking the temperature you realized that it was about 100°C. You decide to let it cool in your dining room with temperature 25°C. The Newton's law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Let T(t) represents the temperature of a bowl of soup set out in the dining room every minute. (i) (ii) (iii) dT dt dT dt dT dt = 100k(T - 25) = k(T – 25) = kT - 100

c. Suppose you are done making a soup. It was too hot and after checking the temperature you realized that it was about 100°C. You decide to let it cool in your dining room with temperature 25°C. The Newton's law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Let T(t) represents the temperature of a bowl of soup set out in the dining room every minute. (i) (ii) (iii) dT dt dT dt dT dt = 100k(T - 25) = k(T – 25) = kT - 100

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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Question

Question 3 Each scenario below describes a situation which can be modeled by a differential equation. Do the following for each scenario: 1. Choose the correct equation. 2. Show and explain your reasoning for your choice.

c. Suppose you are done making a soup. It was too hot and after checking the temperature
you realized that it was about 100°C. You decide to let it cool in your dining room with
temperature 25°C. The Newton's law of Cooling states that the rate of cooling of an object
is proportional to the temperature difference between the object and its surroundings,
provided that this difference is not too large. Let T(t) represents the temperature of a
bowl of soup set out in the dining room every minute.
(i)
(ii)
(iii)
dT
dt
dT
dt
dT
dt
=
100k (T25)
-
= k(T - 25)
= kT-100

a. A biologist is making a bacteria culture starting with 100 cells in a petri dish. The
biologist is recording the number of bacteria every hour. Let N(t) be the number of
bacteria in the petri dish after t hours. After many recording the biologist finds that the
change in the number of bacteria is proportional to the square of the number of bacteria
present at the time of recording.
(i)
dN
dt
€
dN
dt
dN
dt
ds
dt
b. A youtuber decides to analyze her subscribers growth. Let S(t) be the number of sub-
scribers after t months. Using an analysis software, she notices that the change in the
number of subscribers is proportional to the square root of the age (in months) of her
account. Note that when her account was 9 months old her subscribers had increased by
300.
(i)
dS
dt
= k(100 – N)²
ds
dt
= k(N² – 100)
= kN²
=
100√t
3000√t
= k√(t+300)
c. Suppose you are done making a soup. It was too hot and after checking the temperature
you realized that it was about 100°C. You decide to let it cool in your dining room with
temperature 25°C. The Newton's law of Cooling states that the rate of cooling of an object
is proportional to the temperature difference between the object and its surroundings,
provided that this difference is not too large. Let T(t) represents the temperature of a
bowl of soup set out in the dining room every minute.

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