Lesson Worksheet: Exponential Growth and Decay Models Mathematics • 9th Grade

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


A mathematical model predicts that the population of a country, 𝑦 million, will be given by the formula 𝑦=17.1(1.02), where π‘₯ 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


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 π‘Ÿ, the annual rate of interest required. Assume interest is compounded annually.

  • Aο€»1+π‘Ÿ100=12
  • B(1+π‘Ÿ)=12
  • Cο€»π‘Ÿ100=2
  • D(1+π‘Ÿ)=2
  • Eο€»1+π‘Ÿ100=2


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 𝐹, the number of fruit flies expected to be in the population in 𝑑 days’ time.

  • A𝐹=150(3)
  • B𝐹=150(3)
  • C𝐹=150(4)
  • D𝐹=150(3)οŠͺ
  • E𝐹=150(4)


The population of Malawi, in millions, is modeled by the exponential function 𝑃(𝑑)=3.62ο€Ή1.029, where 𝑑 is the time in years since January 1, 1960.

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

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

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

Find the function that represents the same exponential model, but with the input 𝑑 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.

  • A𝑃(𝑑)=11.36ο€½2ο‰ο‘‰οŽ‘οŽ£ο’οŽ‘οŽ€
  • B𝑃(𝑑)=11.36ο€Ή2
  • C𝑃(𝑑)=14.65ο€Ή2
  • D𝑃(𝑑)=ο€Ή2ο…οŠ¨ο
  • E𝑃(𝑑)=11.36ο€½2ο‰ο‘‰οŽ’οŽ₯ο’οŽ§οŽŸ


Let the population of a city be π‘₯. If the population increases by 13% each year, what will the population of the city be in nine years’ time?

  • Alog(π‘₯+0.13)
  • Blog(π‘₯+0.13)
  • C1.17π‘₯
  • D0.286π‘₯
  • E3.004π‘₯


The number of users of a new search engine is increasing every month and can be found using the equation 𝑦=500(1.19), where 𝑦 represents the number of users and π‘₯ represents the number of months since the search engine’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


The number of tourists visiting a theme park increases every year and can be found using the equation 𝑦=1.1(1.045), where 𝑦 million is the number of visitors 𝑑 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?


Rewrite 𝑃(𝑑)=3.62(1.029) in the form 𝑃(𝑑)=𝑃(2)οŠ¦ο‘‰ο‘€, with π‘˜ to two decimal places. What is the significance of the number π‘˜?

  • A𝑃(𝑑)=3.62(2),π‘˜ο‘‰οŽŸο’οŽŸοŽ£οŽ  is the number of years it takes for the population to double
  • B𝑃(𝑑)=3.62(2),π‘˜ο‘‰οŽŸο’οŽŸοŽ£οŽ  is the number of years it takes for the population to triple
  • C𝑃(𝑑)=3.62(2),π‘˜ο‘‰οŽ‘οŽ£ο’οŽ‘οŽ€ is the number of years it takes for the population to double
  • D𝑃(𝑑)=3.62(2),π‘˜ο‘‰οŽ‘οŽ£ο’οŽ‘οŽ€ is the number of years it takes for the population to triple
  • E𝑃(𝑑)=(2),π‘˜ο‘‰οŽ‘οŽ£ο’οŽ‘οŽ€ is the number of years it takes for the population to double


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 𝑑 days later is given by 𝐴=4.3β‹…2ο‘‰οŽ’, when will the algae completely cover the bottom of the swimming pool?

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


Noah’s bank account gives him 5.6% interest on his balance each month. He models his balance after π‘šβ‰₯1 months with the recursive formula π‘Ž=(1+0.056)π‘Žο‰ο‰οŠ±οŠ§. If his initial deposit is 450.00, after how many months will his balance be greater than $600?

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