APPLICATIONS

Growth and Decay - the rate of increase or decrease of a certain

population/substance is directly proportional to

the amount of population/substance present.

*Variable Separable

Let P – population or amount of substance

Example 1: A certain population grows exponentially;

the population grows from 3,500 people to 6245

people in 8 years. Find the population after 20

years and how long it takes for the population to

double? Triple?

Example 2: Bacteria can multiply at an alarming rate

when each bacteria splits into new cells. Thus

doubling. If we start with only one bacterium

which can double every hour, how many bacteria

do we have by the end of the 14th hour? By the

end of the day?

Example 3: The number of bacteria in a yeast culture grows

at a rate which is proportional to the number present.

The bacteria count in a culture doubles in three

hours. At the end of 15 hours, the count is one

million. How many bacteria were in the count

originally?

Example 4: The number of bacteria in a liquid culture is

observed to grow at a rate proportional to the

number of cells present. At the beginning of the

experiment there are 10,000 cells and after three

hours there are 50,000 cells . How many will there

be after one day of growth if this unlimited growth

continues? What is the doubling time (the

amount of time it takes for a given population to

double in size) of the bacteria?

Example 5: Radium decomposes at a rate proportional

to the quantity of radium present. Suppose that it

is found that in 25 years, approximate 11% of a

certain quantity of radium is decomposed,

determine approximately how long it will take

for one-half of the original amount of radium to

decompose?

Example 6: The population of Canada was 24,070,000 in

1980 while in 1990 it was 26,620,000. Assuming the

population is growing according to the principle of

growth with no food or space limitation, (a) find the

time their population will double. For the same years,

the population of Kenya was 16,681,000 and

24,229,000, respectively. (b) Find the population as a

function of time and (c) In what year do the population

of Canada and Kenya become equal?

Example 7: By natural increase, a city whose population is

40000 doubles in 50 years. There is a net addition

of 400 persons per year because of the people

leaving and moving into the city. Estimate its

population in 10 years. (Hint: First find the natural

growth proportionality factor. Assume rate of

increase of population is proportional to the

number of persons in the city at any time t.

Newton’s Law of Cooling - states that the rate of change of a

temperature of a certain body is directly

proportional to the difference in temperature

between the body and the medium/surrounding to

which the body is placed.

*Variable Separable

Let T – temperature of the body

Tm – temperature of the medium/surrounding

Example 8: A body whose temperature is 180C is

immersed in a liquid which is kept at a constant

temperature of 60C. In one minute, the

temperature of the immersed body is 120C. How

long will it take for the body’s temperature to

decrease to 90C?

Example 9: A thermometer reading 18F is brought into a

room the temperature of which is 70F. One

minute later, the reading is 31F. (a) Determine

the temperature reading as a function of time and

in particular, (b) find the temperature reading 5

minutes after the thermometer is first brought into

the room.

Example 10: The temperature of a cup of coffee is initially

at 65.55C. After two minutes it cools to 54.44C. If

the ambient temperature be at 21.11C, (a) find the

equation with respect to time and (b) how long

should I wait for the coffee to cool down to 43.33

degree Celsius?

Example 11: Detective Conan is investigating the death of a

man found stabbed at the back in a restaurant’s

bathroom at 22C. The body was found dead at 24C

(body temperature) at around 2:30PM by a patrol

officer. The man was last seen entering the bathroom

of a restaurant at around 1P.M. the day before he

was found dead. Conan assumed that at the time of

his death, his body temperature drops by 0.2C from

his normal body temperature of 37C. Find the time

of his death.

Example 12: A thermometer reading of 70F is placed in an

oven preheated to a constant temperature. Through a

glass window in the oven door, an observer records

that the thermometer reads 110F over ½ minute

and 145F after 1 minute. How hot is the oven?

Example 13: As part of his summer job at a restaurant, Jim learned to cook

up a big pot of soup late at night, just before closing time (10P.M.), so

that there would be plenty of soup to feed costumers the next day. He

also found out that, while refrigeration was essential to preserve the

soup overnight, the soup was too hot to be put directly into the fridge

when it was ready. (The soup had just boiled at 100C, and the fridge

was not powerful enough to accommodate a big pot of soup if It was

any warmer than 20C). Jim discovered that by cooling the pot in a

sink full of cold water, (kept running, so that its temperature was

roughly constant at 5C) and stirring occasionally, he could bring the

temperature o the soup to 60C in 10 minutes. How long before

closing time should the soup be ready so that Jim could put it in the

fridge and leave on time?

Example 14: A cup of hot coffee is left standing on a table

in a room where the temperature is 75F. The

temperature of the coffee is initially measured to be

186F and after one minute, the temperature

drops to 182F. (a) Find a function of time that best

fits these temperature readings. (b) At what time

does the coffee reach the temperature 83.3244F?

(c) At t=12.25 minutes, what is the temperature

of the coffee?

Example 15: At 1:00 PM a thermometer reading of 100F is

taken out where the room temperature is 40F. At

1:02 PM, the reading is 83F. At 1:05 PM, the

thermometer reading is taken back indoors where

the air is 120F. What is the thermometer reading at

1:09 PM?

Example 16: At 1:00 p.m. a thermometer reading 10F is

removed from a freezer and placed in a room whose

temperature is 65F. At 1:05 p.m., the thermometer

reads 25F. Later, the thermometer is placed back in

the freezer. At 1:30 p.m., the thermometer reads

32F. When was the thermometer returned to the

freezer and what was the thermometer reading this

time?