Worksheet: Exponential Growth and Decay Models

In this worksheet, we will practice modeling exponential growth and decay arising from the differential equation y′=±ky.

Q1:

A mathematical model predicts that the population of a country, ๐‘ฆ million, will be given by the formula ๐‘ฆ = 1 7 . 1 ( 1 . 0 2 ) ๏— , where ๐‘ฅ is the number of years since 2015. Use this model to predict the population of the country, to the nearest million, in both 2021 and 2022.

  • A19 million, 21 million
  • B18 million, 21 million
  • C19 million, 20 million
  • D18 million, 20 million
  • E18 million, 19 million

Q2:

Chloe wants to invest some money. She would like the value of her investment to double in 10 years. Write an equation that can be used to find ๐‘Ÿ , the annual rate of interest required. Assume interest is compounded annually.

  • A ๏€ป 1 + ๐‘Ÿ 1 0 0 ๏‡ = 1 2 ๏Šง ๏Šฆ
  • B ๏€ป ๐‘Ÿ 1 0 0 ๏‡ = 2 ๏Šง ๏Šฆ
  • C ( 1 + ๐‘Ÿ ) = 2 ๏Šง ๏Šฆ
  • D ๏€ป 1 + ๐‘Ÿ 1 0 0 ๏‡ = 2 ๏Šง ๏Šฆ
  • E ( 1 + ๐‘Ÿ ) = 1 2 ๏Šง ๏Šฆ

Q3:

A population of fruit flies quadruples every three days. Today, there were 150 fruit flies in the population under investigation.

Assuming the population continues to grow at the same rate, write an equation that can be used to find ๐น , the number of fruit flies expected to be in the population in ๐‘‘ daysโ€™ time.

  • A ๐น = 1 5 0 ( 4 ) ๏ถ ๏Žข
  • B ๐น = 1 5 0 ( 3 ) ๏ถ ๏Žฃ
  • C ๐น = 1 5 0 ( 4 ) ๏Šฉ ๏Œฝ
  • D ๐น = 1 5 0 ( 3 ) ๏Šช ๏Œฝ
  • E ๐น = 1 5 0 ( 3 ) ๏ถ ๏Žข

Q4:

The population of Malawi, in millions, is modeled by the exponential function ๐‘ƒ ( ๐‘ก ) = 3 . 6 2 ๏€น 1 . 0 2 9 ๏… ๏ , where ๐‘ก is the time in years since January 1 1960.

To the nearest month, how long does it take for the population to double?

  • A 27 years and 2 months
  • B 21 years
  • C 26 years
  • D 21 years and 4 months
  • E 24 years and 3 months

Which year will be the first to start with a population of more than 20 million?

Find the function which represents the same exponential model, but with the input ๐‘ก now being the time in years since January 1 2000. Express this function using a base of 2 rather than the previously used 1.029.

  • A ๐‘ƒ ( ๐‘ก ) = 1 1 . 3 6 ๏€ฝ 2 ๏‰ ๏‘‰ ๏Žข ๏Žฅ ๏’ ๏Žง ๏ŽŸ
  • B ๐‘ƒ ( ๐‘ก ) = 1 4 . 6 5 ๏€น 2 ๏… ๏
  • C ๐‘ƒ ( ๐‘ก ) = 1 1 . 3 6 ๏€ฝ 2 ๏‰ ๏‘‰ ๏Žก ๏Žฃ ๏’ ๏Žก ๏Žค
  • D ๐‘ƒ ( ๐‘ก ) = ๏€น 2 ๏… ๏Šจ ๏
  • E ๐‘ƒ ( ๐‘ก ) = 1 1 . 3 6 ๏€น 2 ๏… ๏

Q5:

Let the population of a city be ๐‘ฅ . If the population increases by 1 3 % each year, what will the population of the city be in nine yearsโ€™ time?

  • A l o g ๏Šฏ ( ๐‘ฅ + 0 . 1 3 )
  • B 0 . 2 8 6 ๐‘ฅ
  • C 1 . 1 7 ๐‘ฅ
  • D 3 . 0 0 4 ๐‘ฅ
  • E l o g ( ๐‘ฅ + 0 . 1 3 )

Q6:

The population of rabbits on a farm grows exponentially. If there are currently 245 rabbits and the relative growth rate is 2 3 % , find a function ๐‘› ( ๐‘ก ) to describe the number of rabbits after ๐‘ก years.

  • A ๐‘› ( ๐‘ก ) = 2 4 5 ๐‘’ ๏Šฆ ๏Ž– ๏Šจ ๏Šฉ
  • B ๐‘› ( ๐‘ก ) = 2 4 5 ( 1 + ๐‘’ ) ๏
  • C ๐‘› ( ๐‘ก ) = 2 4 5 ๐‘’ ๏Šฆ ๏Ž– ๏Šจ ๏Šฉ ๏
  • D ๐‘› ( ๐‘ก ) = 2 4 5 ( 1 + ๐‘’ ) ๏Šฆ ๏Ž– ๏Šจ ๏Šฉ

Q7:

The number of users of a new search engine is increasing every month and can be found using the equation ๐‘ฆ = 5 0 0 ( 1 . 1 9 ) ๏— , where ๐‘ฆ represents the number of users and ๐‘ฅ represents the number of months since the search engineโ€™s launch. If the search engine was launched on the 1st of March, in which month would the search engine have 2 0 0 0 users?

  • ANovember
  • BSeptember
  • COctober
  • DJune
  • EAugust

Q8:

The number of tourists visiting a theme park increases every year and can be found using the equation ๐‘ฆ = 1 . 1 ( 1 . 0 4 5 ) ๏ , where ๐‘ฆ million is the number of visitors ๐‘ก years after 2010. If the number of visitors continues to increase at the same rate, in what year will the park first reach 2 million visitors?

Q9:

Rewrite ๐‘ƒ ( ๐‘ก ) = 3 . 6 2 ( 1 . 0 2 9 ) ๏ in the form ๐‘ƒ ( ๐‘ก ) = ๐‘ƒ ( 2 ) ๏Šฆ ๏‘‰ ๏‘€ , with ๐‘˜ to two decimal places. What is the significance of the number ๐‘˜ ?

  • A ๐‘ƒ ( ๐‘ก ) = 3 . 6 2 ( 2 ) , ๐‘˜ ๏‘‰ ๏Žก ๏Žฃ ๏’ ๏Žก ๏Žค is the number of years it takes for the population to double
  • B ๐‘ƒ ( ๐‘ก ) = 3 . 6 2 ( 2 ) , ๐‘˜ ๏‘‰ ๏Žก ๏Žฃ ๏’ ๏Žก ๏Žค is the number of years it takes for the population to triple
  • C ๐‘ƒ ( ๐‘ก ) = 3 . 6 2 ( 2 ) , ๐‘˜ ๏‘‰ ๏ŽŸ ๏’ ๏ŽŸ ๏Žฃ ๏Ž  is the number of years it takes for the population to triple
  • D ๐‘ƒ ( ๐‘ก ) = 3 . 6 2 ( 2 ) , ๐‘˜ ๏‘‰ ๏ŽŸ ๏’ ๏ŽŸ ๏Žฃ ๏Ž  is the number of years it takes for the population to double
  • E ๐‘ƒ ( ๐‘ก ) = ( 2 ) , ๐‘˜ ๏‘‰ ๏Žก ๏Žฃ ๏’ ๏Žก ๏Žค is the number of years it takes for the population to double

Q10:

On July 5, green algae was found on the bottom of a swimming pool whose width is 6 m and length is 12 m. If the area, in mm2, the algae covers ๐‘ก days later is given by ๐ด = 4 . 3 โ‹… 2 ๏‘‰ ๏Žข , when will the algae completely cover the bottom of the swimming pool?

  • AJuly 18
  • BAugust 22
  • CJuly 15
  • DSeptember 15
  • EAugust 18

Q11:

Anthonyโ€™s bank account gives him 5 . 6 % interest on his balance each month. He models his balance after ๐‘š โ‰ฅ 1 months with the recursive formula ๐‘Ž = ( 1 + 0 . 0 5 6 ) ๐‘Ž ๏‰ ๏‰ ๏Šฑ ๏Šง . If his initial deposit is 450.00, after how many months will his balance be greater than $600?

Q12:

The function ๐‘ƒ ( ๐‘ก ) = ๐ด ๐‘ ๏ represents a population, in millions, ๐‘ก years after 1,970 that is growing at an annual rate of 3 . 5 % and started at 13.2 million in 1,970. What is the value of ๐‘ ?

Q13:

The value of a car depreciates at a rate of 1 5 % each year.

Write an equation that can be used to calculate ๐‘‰ , the value of a car, in dollars, ๐‘ก years after it was purchased for ๐ถ d o l l a r s .

  • A ๐‘‰ = ๐ถ ( 0 . 7 5 ) ๏
  • B ๐‘‰ = ๐ถ ( 0 . 8 5 ) ๏
  • C ๐‘‰ = ๐ถ ( 1 . 1 5 ) ๏
  • D ๐‘‰ = ๐ถ ( 1 . 8 5 ) ๏
  • E ๐‘‰ = ๐ถ ( 0 . 1 5 ) ๏

What is the total depreciation in the carโ€™s value over 6 years? Give your answer to the nearest percent.

Q14:

The number of cars worldwide, ๐‘ก years after 2015, can be modeled by the formula ๐‘ = 1 0 โ‹… ๐‘’ ๏Šฏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šจ ๏Šญ ๏Šญ ๏ . In what year will there be 1.4 billion cars worldwide?

  • A2027
  • B2037
  • C2066
  • D2065
  • E2029

Q15:

Let ๐น ( ๐‘ก ) be the population of bacteria in a culture. At time ๐‘ก = 0 , the population is 4 million. Suppose that, after ๐‘ก hours, the bacteria grow at the instantaneous rate of change of 3 ๏ million bacteria per hour. Estimate the number of bacteria at time ๐‘ก = 1 in millions to two decimal places.

  • A6.73 million
  • B4.82 million
  • C2.18 million
  • D5.82 million
  • E3.59 million

Q16:

The rate at which a radioactive substance decays is proportional to the remaining number of atoms. The differential equation which can be used to describe this process follows d d ๐‘ ๐‘ก = โˆ’ ๐œ† ๐‘ , where ๐‘ represents the number of atoms remaining after ๐‘ก seconds. The proportionality constant ๐œ† is considered the decay constant for this process. If ๐‘ ๏Šฆ represents the number of remaining atoms at ๐‘ก = 0 seconds, find the general solution.

  • A ๐‘ = ๐‘ ๐‘’ ๏Šฆ ๏ฟ ๏
  • B ๐‘ = ๐‘ ๐‘’ ๏Šฆ ๏‘‰ ๏‘ธ
  • C ๐‘ = ๐‘ ๐‘’ ๏Šฆ ๏Šฑ ๏ฟ ๏
  • D ๐‘ = ๐‘ ๐‘’ ๏Šฆ ๏Šฑ ๏‘‰ ๏‘ธ

Q17:

Hannah and Mason are playing a game where they roll 6-sided dice, and then they remove all the dice showing a 1. Then they roll the remaining dice and remove all the dice showing a 1 again, and so on.

Hannah and Mason started with 42 dice. According to the law of probability, find an explicit formula for the number of dice remaining after ๐‘Ÿ rounds of the game.

  • A ๐‘ = 6 โ‹… ๏€ผ 4 1 4 2 ๏ˆ ๏Ž
  • B ๐‘ = 4 2 โ‹… ๏€ผ 5 6 ๏ˆ ๏Ž
  • C ๐‘ = 6 โ‹… ๏€ผ 1 4 2 ๏ˆ ๏Ž
  • D ๐‘ = 4 2 โ‹… ๏€ผ 1 6 ๏ˆ ๏Ž
  • E ๐‘ = 4 2 โ‹… ๏€ผ 5 6 ๏ˆ ๐‘› ๏Ž

How many rounds does it take to remove roughly 2 3 of the dice?

Q18:

The black rhino is an endangered species. Its global population has fallen from 6 5 0 0 0 in 1970 to 2 3 0 0 in 1993. By modeling the decline as exponential, answer the following questions.

Write an equation in the form ๐‘ƒ = ๐‘Ž ๐‘ ๏ , where ๐‘ƒ is the population of black rhinos ๐‘ก years after 1970. Round your values of ๐‘Ž and ๐‘ to 3 decimal places if necessary.

  • A ๐‘ƒ = 5 6 2 2 5 ๏
  • B ๐‘ƒ = 6 5 0 0 0 ( 1 . 8 6 5 ) ๏
  • C ๐‘ƒ = 6 5 0 0 0 ( ๐‘ก ) ๏Šง ๏Ž– ๏Šฎ ๏Šฌ ๏Šซ
  • D ๐‘ƒ = 6 5 0 0 0 ( 0 . 8 6 5 ) ๏
  • E ๐‘ƒ = 6 5 0 0 0 ( ๐‘ก ) ๏Šฆ ๏Ž– ๏Šฎ ๏Šฌ ๏Šซ

According to the model, what was the black rhino population in 1980?

According to the model, by how much did the black rhino population fall between 1980 and 1990?

Q19:

A doctor injected a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there were 4.75 milligrams of dye remaining in the patientโ€™s system. Which of the following is an appropriate model for this situation?

  • A ๐‘“ ( ๐‘ก ) = 4 . 7 5 1 + 1 3 ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฎ ๏Šฉ ๏Šฏ ๏Šจ ๏Šซ ๏
  • B ๐‘“ ( ๐‘ก ) = 1 3 ( 0 . 0 8 0 5 ) ๏
  • C ๐‘“ ( ๐‘ก ) = 1 3 ๐‘’ ( ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏Šฉ ๏Šฏ ๏ )
  • D ๐‘“ ( ๐‘ก ) = 1 3 ๐‘’ ๏Šฆ ๏Ž– ๏Šฏ ๏Šง ๏Šฏ ๏Šซ ๏

Q20:

The results of a medical study showed that, in healthy adults, the half-life of caffeine is 5.7 hours. So, if an adult consumes 250 mg of caffeine in their breakfast coffee at 6 am, they will have approximately 125 mg of caffeine in their system at 11:40 am.

If a person drinks a can of cola containing 30 mg of caffeine, the amount of caffeine, ๐ถ , in their system ๐‘ก hours later can be found using the equation ๐ถ = 3 0 ๏€ผ 1 2 ๏ˆ ( ) ๏‘‰ ๏Žค ๏’ ๏Žฆ .

Write the equation in the form ๐ถ = ๐ด ( ๐‘ ) ๏ , giving values to 3 decimal places if necessary.

  • A ๐ถ = 1 5 ( 0 . 1 7 5 ) ๏
  • B ๐ถ = 3 0 ( 0 . 0 1 9 ) ๏
  • C ๐ถ = 3 0 ( 2 . 3 8 7 ) ๏
  • D ๐ถ = 3 0 ( 0 . 8 8 5 ) ๏
  • E ๐ถ = 1 5 ( 5 . 7 ) ๏

Q21:

The number of marine organisms in a pool, ๐‘ฆ , after ๐‘› weeks is given by the formula ๐‘ฆ = 5 , 6 3 1 ๏€ผ 1 2 ๏ˆ ๏Š ๏Šฑ ๏Šง . How many marine organisms will there be in the pool after 4 weeks? Give your answer to the nearest whole number.

Q22:

At the start of an experiment, a scientist has a sample which contains 250 milligrams of a radioactive isotope. The radioactive isotope decays exponentially, so that after 250 minutes there are only 32.0 milligrams of the isotope left.

Write the mass of isotope in milligrams, ๐‘€ , as a function of the time in minutes, ๐‘ก , since the experiment started. Give your answer in the form ๐‘€ ( ๐‘ก ) = ๐ด ๐‘’ ๏Œป ๏ , rounding ๐ด and ๐‘ to three significant figures.

  • A ๐‘€ ( ๐‘ก ) = 2 5 0 ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฆ ๏Šญ ๏Šฉ ๏Šช ๏
  • B ๐‘€ ( ๐‘ก ) = 2 5 0 ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฆ ๏Šฎ ๏Šจ ๏Šฉ ๏
  • C ๐‘€ ( ๐‘ก ) = 3 2 ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฆ ๏Šญ ๏Šฉ ๏Šช ๏
  • D ๐‘€ ( ๐‘ก ) = 2 5 0 ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฆ ๏Šซ ๏Šฌ ๏Šญ ๏
  • E ๐‘€ ( ๐‘ก ) = 3 2 ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฆ ๏Šฏ ๏Šง ๏Šช ๏

Find the half-life of the isotope, giving your answer to the nearest minute.

Q23:

Carbon dating calculates the amount of the isotope carbon-14 that was fixed from the atmosphere when an animal died and stopped absorbing it. The isotopeโ€™s quantity is then reduced by half every 5 7 3 0 years. Let the amount of the isotope after ๐‘ก years be ๐ด ( ๐‘ก ) .

Write an equation relating ๐ด ( ๐‘ก ) to ๐ด ( ๐‘ก + 5 7 3 0 ) .

  • A ๐ด ( ๐‘ก ) = ๏€ผ 1 2 ๏ˆ ๐ด ( ๐‘ก + 5 7 3 0 )
  • B ๐ด ( ๐‘ก ) = ๏€ผ 1 4 ๏ˆ ๐ด ( ๐‘ก + 5 7 3 0 )
  • C ๐ด ( ๐‘ก + 5 7 3 0 ) = ๏€ผ 1 3 ๏ˆ ๐ด ( ๐‘ก )
  • D ๐ด ( ๐‘ก + 5 7 3 0 ) = ๏€ผ 1 4 ๏ˆ ๐ด ( ๐‘ก )
  • E ๐ด ( ๐‘ก + 5 7 3 0 ) = ๏€ผ 1 2 ๏ˆ ๐ด ( ๐‘ก )

Write expressions for ๐ด ( ๐‘ก + 1 1 4 6 0 ) and ๐ด ( ๐‘ก + 1 7 1 9 0 ) .

  • A ๐ด ( ๐‘ก ) = ๏€ผ 1 4 ๏ˆ ๐ด ( ๐‘ก ) , ๐ด ( ๐‘ก ) = ๏€ผ 1 8 ๏ˆ ๐ด ( ๐‘ก + 1 7 1 9 0 )
  • B ๐ด ( ๐‘ก + 1 1 4 6 0 ) = ๏€ผ 1 3 ๏ˆ ๐ด ( ๐‘ก ) , ๐ด ( ๐‘ก + 1 7 1 9 0 ) = ๏€ผ 1 9 ๏ˆ ๐ด ( ๐‘ก )
  • C ๐ด ( ๐‘ก + 1 1 4 6 0 ) = ๏€ผ 1 2 ๏ˆ ๐ด ( ๐‘ก ) , ๐ด ( ๐‘ก + 1 7 1 9 0 ) = ๏€ผ 1 7 ๏ˆ ๐ด ( ๐‘ก )
  • D ๐ด ( ๐‘ก ) = ๏€ผ 1 2 ๏ˆ ๐ด ( ๐‘ก ) , ๐ด ( ๐‘ก ) = ๏€ผ 1 9 ๏ˆ ๐ด ( ๐‘ก + 1 7 1 9 0 )
  • E ๐ด ( ๐‘ก + 1 1 4 6 0 ) = ๏€ผ 1 4 ๏ˆ ๐ด ( ๐‘ก ) , ๐ด ( ๐‘ก + 1 7 1 9 0 ) = ๏€ผ 1 8 ๏ˆ ๐ด ( ๐‘ก )

Write a formula relating ๐ด ( ๐‘ก + 5 7 3 0 ๐‘› ) to ๐ด ( ๐‘ก ) for a positive integer ๐‘› .

  • A ๐ด ( ๐‘ก ) = ๏€ป ๐‘› 2 ๏‡ ๐ด ( 2 ๐‘ก + 5 7 3 0 ๐‘› )
  • B ๐ด ( ๐‘ก + 5 7 3 0 ๐‘› ) = ๏€ผ 1 2 ๏ˆ ๐ด ( ๐‘ก ) ๏Š
  • C ๐ด ( ๐‘ก + 5 7 3 0 ๐‘› ) = ๏€ผ 1 4 ๏ˆ ๐ด ( ๐‘ก ) ๏Š
  • D ๐ด ( ๐‘ก ) = ๏€ป ๐‘› 2 ๏‡ ๐ด ( ๐‘ก + 5 7 3 0 ๐‘› )
  • E ๐ด ( ๐‘ก + 5 7 3 0 ๐‘› ) = ๏€ผ 1 3 ๏ˆ ๐ด ( ๐‘ก ) ๏Š

Assume that, every 5 7 3 0 2 years, the carbon-14 isotope is reduced by the same ratio ๐‘Ÿ . By writing 5 7 3 0 as 5 7 3 0 2 + 5 7 3 0 2 , what is ๐‘Ÿ ?

  • A ๐‘Ÿ = 1 4
  • B ๐‘Ÿ = ๏„ž 1 3
  • C ๐‘Ÿ = ๏„ž 1 2
  • D ๐‘Ÿ = 1 2
  • E ๐‘Ÿ = 1 3

Q24:

Every day, following treatment with a weed killer, the area of clover in a garden is reduced to one-third of the previous dayโ€™s area. On the day the weed killer was applied, there was approximately 40 m2 of clover in the garden. Write an equation that can be used to find ๐ถ , the area of clover in the garden after ๐‘ก days.

  • A ๐ถ = 4 0 ( ๐‘ก ) ๏Ž  ๏Žข
  • B ๐ถ = 1 3 ( ๐‘ก ) ๏Šช ๏Šฆ
  • C ๐ถ = ๏€ผ 4 0 3 ๏ˆ ๏
  • D ๐ถ = 1 3 ( 4 0 ) ๏
  • E ๐ถ = 4 0 ๏€ผ 1 3 ๏ˆ ๏

Q25:

The concentration of aspirin in human blood ๐‘ก hours after the intake of a normal dose ๐‘ ๏Šฆ can be modeled by the function ๐‘ = ๐‘ ๏€ผ 1 2 ๏ˆ ๏Šฆ ๏‘‰ ๏Žข .

What is the half-life of aspirin, that is the time it takes for half of the initial dose to be eliminated?

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