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=17.1(1.02)๏—, 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.

  • A18 million, 21 million
  • B19 million, 21 million
  • C18 million, 19 million
  • D19 million, 20 million
  • E18 million, 20 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๏€ป1+๐‘Ÿ100๏‡=12๏Šง๏Šฆ
  • B(1+๐‘Ÿ)=12๏Šง๏Šฆ
  • C๏€ป๐‘Ÿ100๏‡=2๏Šง๏Šฆ
  • D(1+๐‘Ÿ)=2๏Šง๏Šฆ
  • E๏€ป1+๐‘Ÿ100๏‡=2๏Šง๏Šฆ

Q3:

A population of fruit flies quadruples every 3 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.

  • A๐น=150(3)๏ถ๏Žข
  • B๐น=150(3)๏ถ๏Žฃ
  • C๐น=150(4)๏ถ๏Žข
  • D๐น=150(3)๏Šช๏Œฝ
  • E๐น=150(4)๏Šฉ๏Œฝ

Q4:

The population of Malawi, in millions, is modeled by the exponential function ๐‘ƒ(๐‘ก)=3.62๏€น1.029๏…๏, where ๐‘ก is the time in years since January 11960.

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

  • A24 years and 3 months
  • B21 years and 4 months
  • C26 years
  • D27 years and 2 months
  • E21 years

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 12000. Express this function using a base of 2 rather than the previously used 1.029.

  • A๐‘ƒ(๐‘ก)=11.36๏€ฝ2๏‰๏‘‰๏Žก๏Žฃ๏’๏Žก๏Žค
  • B๐‘ƒ(๐‘ก)=11.36๏€น2๏…๏
  • C๐‘ƒ(๐‘ก)=14.65๏€น2๏…๏
  • D๐‘ƒ(๐‘ก)=๏€น2๏…๏Šจ๏
  • E๐‘ƒ(๐‘ก)=11.36๏€ฝ2๏‰๏‘‰๏Žข๏Žฅ๏’๏Žง๏ŽŸ

Q5:

Let the population of a city be ๐‘ฅ. If the population increases by 13% each year, what will the population of the city be in nine yearsโ€™ time?

  • Alog(๐‘ฅ+0.13)
  • Blog๏Šฏ(๐‘ฅ+0.13)
  • C1.17๐‘ฅ
  • D0.286๐‘ฅ
  • E3.004๐‘ฅ

Q6:

The number of users of a new search engine is increasing every month and can be found using the equation ๐‘ฆ=500(1.19)๏—, 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 2,000 users?

  • AOctober
  • BSeptember
  • CAugust
  • DNovember
  • EJune

Q7:

The number of tourists visiting a theme park increases every year and can be found using the equation ๐‘ฆ=1.1(1.045)๏, 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?

Q8:

Rewrite ๐‘ƒ(๐‘ก)=3.62(1.029)๏ 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‘˜?

  • A๐‘ƒ(๐‘ก)=3.62(2),๐‘˜๏‘‰๏ŽŸ๏’๏ŽŸ๏Žฃ๏Ž  is the number of years it takes for the population to double
  • B๐‘ƒ(๐‘ก)=3.62(2),๐‘˜๏‘‰๏ŽŸ๏’๏ŽŸ๏Žฃ๏Ž  is the number of years it takes for the population to triple
  • C๐‘ƒ(๐‘ก)=3.62(2),๐‘˜๏‘‰๏Žก๏Žฃ๏’๏Žก๏Žค is the number of years it takes for the population to double
  • D๐‘ƒ(๐‘ก)=3.62(2),๐‘˜๏‘‰๏Žก๏Žฃ๏’๏Žก๏Žค is the number of years it takes for the population to triple
  • E๐‘ƒ(๐‘ก)=(2),๐‘˜๏‘‰๏Žก๏Žฃ๏’๏Žก๏Žค is the number of years it takes for the population to double

Q9:

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=4.3โ‹…2๏‘‰๏Žข, when will the algae completely cover the bottom of the swimming pool?

  • AJuly 15
  • BSeptember 15
  • CAugust 22
  • DAugust 18
  • EJuly 18

Q10:

Noahโ€™s bank account gives him 5.6% interest on his balance each month. He models his balance after ๐‘šโ‰ฅ1 months with the recursive formula ๐‘Ž=(1+0.056)๐‘Ž๏‰๏‰๏Šฑ๏Šง. If his initial deposit is 450.00, after how many months will his balance be greater than $600?

Q11:

The function ๐‘ƒ(๐‘ก)=๐ด๐‘๏ 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‘?

Q12:

The value of a car depreciates at a rate of 15% 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ถdollars.

  • A๐‘‰=๐ถ(0.75)๏
  • B๐‘‰=๐ถ(0.85)๏
  • C๐‘‰=๐ถ(1.85)๏
  • D๐‘‰=๐ถ(1.15)๏
  • E๐‘‰=๐ถ(0.15)๏

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

Q13:

The number of cars worldwide, ๐‘ก years after 2015, can be modeled by the formula ๐‘=10โ‹…๐‘’๏Šฏ๏Šฆ๏Ž–๏Šฆ๏Šจ๏Šญ๏Šญ๏. In what year will there be 1.4 billion cars worldwide?

  • A2066
  • B2029
  • C2027
  • D2065
  • E2037

Q14:

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.

Q15:

Scarlett 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.

Scarlett 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.

  • A๐‘=6โ‹…๏€ผ4142๏ˆ๏Ž
  • B๐‘=42โ‹…๏€ผ56๏ˆ๐‘›๏Ž
  • C๐‘=6โ‹…๏€ผ142๏ˆ๏Ž
  • D๐‘=42โ‹…๏€ผ16๏ˆ๏Ž
  • E๐‘=42โ‹…๏€ผ56๏ˆ๏Ž

How many rounds does it take to remove roughly 23 of the dice?

Q16:

The black rhino is an endangered species. Its global population has fallen from 65,000 in 1970 to 2,300 in 1993. By modeling the decline as exponential, answer the following questions.

Write an equation in the form ๐‘ƒ=๐‘Ž๐‘๏, 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.

  • A๐‘ƒ=65,000(๐‘ก)๏Šง๏Ž–๏Šฎ๏Šฌ๏Šซ
  • B๐‘ƒ=65,000(1.865)๏
  • C๐‘ƒ=56,225๏
  • D๐‘ƒ=65,000(๐‘ก)๏Šฆ๏Ž–๏Šฎ๏Šฌ๏Šซ
  • E๐‘ƒ=65,000(0.865)๏

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?

Q17:

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=30๏€ผ12๏ˆ๏€ป๏‡๏‘‰๏Žค๏’๏Žฆ.

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

  • A๐ถ=30(0.885)๏
  • B๐ถ=30(2.387)๏
  • C๐ถ=30(0.019)๏
  • D๐ถ=15(0.175)๏
  • E๐ถ=15(5.7)๏

Q18:

The number of marine organisms in a pool, ๐‘ฆ, after ๐‘› weeks is given by the formula ๐‘ฆ=5,631๏€ผ12๏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.

Q19:

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 5,730 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+5,730).

  • A๐ด(๐‘ก+5,730)=๏€ผ13๏ˆ๐ด(๐‘ก)
  • B๐ด(๐‘ก+5,730)=๏€ผ12๏ˆ๐ด(๐‘ก)
  • C๐ด(๐‘ก)=๏€ผ14๏ˆ๐ด(๐‘ก+5,730)
  • D๐ด(๐‘ก)=๏€ผ12๏ˆ๐ด(๐‘ก+5,730)
  • E๐ด(๐‘ก+5,730)=๏€ผ14๏ˆ๐ด(๐‘ก)

Write expressions for ๐ด(๐‘ก+11,460) and ๐ด(๐‘ก+17,190).

  • A๐ด(๐‘ก+11,460)=๏€ผ14๏ˆ๐ด(๐‘ก),๐ด(๐‘ก+17,190)=๏€ผ18๏ˆ๐ด(๐‘ก)
  • B๐ด(๐‘ก+11,460)=๏€ผ13๏ˆ๐ด(๐‘ก),๐ด(๐‘ก+17,190)=๏€ผ19๏ˆ๐ด(๐‘ก)
  • C๐ด(๐‘ก)=๏€ผ12๏ˆ๐ด(๐‘ก),๐ด(๐‘ก)=๏€ผ19๏ˆ๐ด(๐‘ก+17,190)
  • D๐ด(๐‘ก+11,460)=๏€ผ12๏ˆ๐ด(๐‘ก),๐ด(๐‘ก+17,190)=๏€ผ17๏ˆ๐ด(๐‘ก)
  • E๐ด(๐‘ก)=๏€ผ14๏ˆ๐ด(๐‘ก),๐ด(๐‘ก)=๏€ผ18๏ˆ๐ด(๐‘ก+17,190)

Write a formula relating ๐ด(๐‘ก+5,730๐‘›) to ๐ด(๐‘ก) for a positive integer ๐‘›.

  • A๐ด(๐‘ก)=๏€ป๐‘›2๏‡๐ด(๐‘ก+5,730๐‘›)
  • B๐ด(๐‘ก)=๏€ป๐‘›2๏‡๐ด(2๐‘ก+5,730๐‘›)
  • C๐ด(๐‘ก+5,730๐‘›)=๏€ผ13๏ˆ๐ด(๐‘ก)๏Š
  • D๐ด(๐‘ก+5,730๐‘›)=๏€ผ14๏ˆ๐ด(๐‘ก)๏Š
  • E๐ด(๐‘ก+5,730๐‘›)=๏€ผ12๏ˆ๐ด(๐‘ก)๏Š

Assume that, every 5,7302 years, the carbon-14 isotope is reduced by the same ratio ๐‘Ÿ. By writing 5,730 as 5,7302+5,7302, what is ๐‘Ÿ?

  • A๐‘Ÿ=12
  • B๐‘Ÿ=14
  • C๐‘Ÿ=๏„ž13
  • D๐‘Ÿ=13
  • E๐‘Ÿ=๏„ž12

Q20:

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.

  • A๐ถ=๏€ผ403๏ˆ๏
  • B๐ถ=13(๐‘ก)๏Šช๏Šฆ
  • C๐ถ=40(๐‘ก)๏Ž ๏Žข
  • D๐ถ=40๏€ผ13๏ˆ๏
  • E๐ถ=13(40)๏

Q21:

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‘๏€ผ12๏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?

Q22:

The population growth rate of bacteria under different conditions is being investigated in a laboratory.

The population of bacteria in Isabellaโ€™s experiment doubles every day, and the population of Jacobโ€™s fruit flies triples every day.

Isabella starts with 81 bacteria. She uses the function ๐‘“(๐‘ก)=81(2)๏ to model the population of bacteria after ๐‘ก days.

Jacob starts with ๐ต flies. Write the function, ๐‘”, Jacob uses to model his fly population after ๐‘ก days.

  • A๐‘”(๐‘ก)=(3)๏
  • B๐‘”(๐‘ก)=๐ต(3)๏
  • C๐‘”(๐‘ก)=3(๐ต)๏
  • D๐‘”(๐‘ก)=๐ต๏
  • E๐‘”(๐‘ก)=(3๐ต)๏

If the two populations are the same after 4 days, how many flies were there in Jacobโ€™s starting population?

Q23:

In a laboratory, a bacteria population quadruples every hour. The population was first measured to be 50 bacteria. Write an equation that can be used to find ๐ต, the bacteria population after ๐‘ก hours.

  • A๐ต=50(๐‘ก)๏Šช
  • B๐ต=4(50)๏
  • C๐ต=4(๐‘ก)๏Šซ๏Šฆ
  • D๐ต=50(4)๏
  • E๐ต=100๏

Q24:

When caffeine is metabolized by our body (that is, when our body breaks down, uses, and absorbs caffeine), the decreasing quantity of caffeine can be modeled by the function ๐‘„=๐‘„๐‘’๏Šฆ๏Šฑ๏Šฆ๏Ž–๏Šง๏, where ๐‘ก is the number of hours after an intake of ๐‘„๏Šฆ. What is the half-life of caffeine in our body? In other words, how long does it take for our body to break down half of the caffeine? Round your answer to the nearest hour.

Q25:

A population of fruit flies quadruples every 3 days.

Write an equation that could be used to calculate ๐‘ƒ, the number of fruit flies after ๐‘ก days, if the initial population was 400.

  • A๐‘ƒ=400(4)๏Šฉ๏
  • B๐‘ƒ=400(๐‘ก)๏Žฃ๏Žข
  • C๐‘ƒ=400๏€ผ๐‘ก3๏ˆ๏Šช
  • D๐‘ƒ=400(3)๏‘‰๏Žฃ
  • E๐‘ƒ=400(4)๏‘‰๏Žข

How many fruit flies will there be after 5 days?

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