Lesson Worksheet: Exponential Growth and Decay Mathematics

In this worksheet, we will practice setting up and solving exponential growth and decay equations and interpreting their solutions.

Q1:

Is the exponential function ๐‘ฆ=4(1.21)๏— growing or decaying?

  • Adecaying
  • Bgrowing

Q2:

A population of bacteria in a petri dish ๐‘ก hours after the culture has started is given by ๐‘ƒ=2,400โ‹…๐‘’๏Šฆ๏Ž–๏Šฆ๏Šฎ๏Šช๏. Amelia says this means that the growth rate is 8.4% per hour. Her friend Elizabeth, however, says that the hourly growth rate is 8.76%. Who is right?

  • AElizabeth
  • BAmelia

Q3:

The given figure shows the concentration ๐‘, in micrograms per liter, of a certain drug in human blood plasma measured at different times. Considering that the concentration after โ„Ž hours can be modeled with the function ๐‘=18โ‹…0.75๏‚, by what percentage does the drugโ€™s concentration decrease every hour?

Q4:

A carโ€™s value depreciates by ๐‘Ÿ% every year. A new car costs ๐‘ƒdollars.

Write a function that can be used to calculate ๐‘‰, the carโ€™s value in dollars, after ๐‘ก years.

  • A๐‘‰(๐‘ก)=๐‘ƒ๏€ป1โˆ’๐‘Ÿ100๏‡๏
  • B๐‘‰(๐‘ก)=๏€ป๐‘ƒโˆ’๐‘Ÿ100๏‡๏
  • C๐‘‰(๐‘ก)=๐‘ƒ๏€ป๐‘Ÿ100๏‡๏
  • D๐‘‰(๐‘ก)=๐‘ƒ๏€ป1+๐‘Ÿ100๏‡๏
  • E๐‘‰(๐‘ก)=๏€ป1โˆ’๐‘Ÿ100๏‡๏

What is the value of ๐‘Ÿ for which the carโ€™s value will be halved in 3 years? Give your answer to the nearest whole number.

Q5:

A man invested 200,000 LE in a project. Each year his investment grows by 9%. Determine the value of his investment after 7 years, giving your answer to the nearest pound.

Q6:

A scientist is considering two termite species: ๐ด and ๐ต. At the start of the experiment, there are 1,233 of ๐ด and 1,640 of ๐ต. They both increase exponentially: the smaller group ๐ด at 1.3% per day, which is higher than ๐ตโ€™s 0.4%. The scientist believes that, despite the fact that ๐ต had a head start, ๐ด will eventually surpass ๐ต in terms of population given its higher rate. She also believes that this will happen by day 30. Is her estimate correct? To use the model, you must round to the nearest integer.

  • ANo
  • BYes

Q7:

After winning in a contest, you are rewarded with either 100,000 regular gold coins or a magical coin that doubles in value every day. The magical coin is worth 1 gold coin on the first day and then doubles in value for 20 days. Which prize would give you a greater number of gold coins at the end of the 20 days?

  • AThe magic coin
  • BThe 100,000 regular gold coins

Q8:

A population grew from 3.62 million to 4.604 million in ten years. What is the annual percentage growth rate of this population? Give your answer to 2 decimal places.

  • A2.43%
  • B12.72%
  • C1.12%
  • D2.72%
  • E7.86%

Q9:

Two cars are bought at the same time. One of them costs $27,000 and depreciates at 7% each year. If the second car costs $39,000, at what rate does it depreciate if they have the same value after 5 years? Give your answer to one decimal place.

Q10:

Consider the decay function ๐‘ฆ=59,0493๏—.

Write the decay function in the form ๐‘ฆ=๐ด(1โˆ’๐‘)๏—, writing the value of ๐‘ accurate to three decimal places.

  • A๐‘ฆ=59,049(1โˆ’0.200)๏—
  • B๐‘ฆ=59,049(1โˆ’0.667)๏—
  • C๐‘ฆ=59,049(1โˆ’0.800)๏—
  • D๐‘ฆ=59,049(1โˆ’0.400)๏—
  • E๐‘ฆ=59,049(1โˆ’0.333)๏—

State the rate of decay as a percent to the nearest tenth.

  • A20.0%
  • B33.3%
  • C40.0%
  • D88.0%
  • E66.7%

Q11:

The production of a gold mine is 4,945 kg per year. A mathematical model predicts that yearly production will go down by 13% every year. What does the model predict the production will be in 7 years time? Give your answer correct to two decimal places.

Q12:

Two vehicles are bought in the same year. One car costs $27,000 and depreciates at a rate of 6% per year. The second car is more expensive but depreciates faster, at 10% per year. How much must this car cost if the two have exactly the same value after 4 years? Give your answer to the nearest dollar.

Q13:

In a laboratory experiment to test a new antibiotic, the population of a colony of bacteria declines by one-third every 6 hours since the beginning of the treatment.

If the initial population was 2,500 bacteria, how many would there be after 10 hours? Give your answer to the nearest 100 bacteria.

Q14:

A cereal manufacturer decides to make their products healthier by reducing the amount of sugar in them. Their target is to reduce the amount of sugar in their product range by 20%. They plan to achieve their target in 4 years.

Write an equation they could use to find ๐‘Ÿ, the annual sugar reduction rate required to achieve their target.

  • A๏€ผ100โˆ’๐‘Ÿ100๏ˆ=0.8๏Šช
  • B(1โˆ’๐‘Ÿ)=0.8๏Šช
  • C๏€ป๐‘Ÿ100๏‡=0.8๏Šช
  • D๏€ป1โˆ’๐‘Ÿ100๏‡=0.2๏Šช
  • E(1โˆ’๐‘Ÿ)=0.2๏Šช

Q15:

After every match a football team plays, they lose 7% of their fans. Before the first match of the championship, they had 83,900 fans. How many fans did they have after the ninth match? Give your answer to the nearest whole number.

Q16:

In 2012, Jacob and Anthony bought cars. The car that Jacob bought cost 27,000 USD. Anthonyโ€™s car cost him 39,000 USD. The table shows the values of these cars in 2012 and 2013.

JacobAnthony
201227,00039,000
201325,38030,225

By what percentage was the price of each car reduced from 2012 to 2013?

  • A6% and 22.5%
  • B6% and 77.5%
  • C94% and 77.5%
  • D94% and 22.5%
  • E11.7% and 34.5%

Fill in the third row if the values were reduced by the same percentage between 2013 and 2014. Give your answers to the nearest dollar.

  • A27,186 and 29,473
  • B23,857 and 23,424
  • C22,628 and 24,148
  • D26,351 and 25,457
  • E21,245 and 22,637

Q17:

An isotope decays with a half-life of 50 years. What is the percentage of decay each year? Give your answer to three decimal places.

  • A5.645%
  • B98.623%
  • C1.377%
  • D3.912%
  • E7.824%

Q18:

At the end of 2000, the population of a country was 22.4 million. Since then, the population has increased by 5.6% every year. What is the population, rounded to the nearest tenth, of the country at the end of 2037?

  • A168.2 million
  • B23.7 million
  • C22.5 million
  • D21.1 million
  • E2.7 million

Q19:

Ethan has 73 rabbits. He believes that he will have ๐‘ง=73(4.21)๏‘ƒ๏Žข rabbits after ๐‘› months. How many rabbits does he expect to have 2 months from now?

Q20:

A cattle farm has 25 cows. The farmer predicts that each year he will have 19% more cows than the year before. How many cows, to the nearest whole number, will he have after 7 years?

Q21:

An area covered in green algae was found on July 5 on the bottom of a swimming pool. The area, in square millimeters, the algae covers ๐‘ก days later is given by ๐ด=1.2โ‹…2๏‘‰๏Žข.

What does 1.2 represent?

  • AIt is the time to reach the bottom of the swimming pool.
  • BIt is the area in square millimeters of the swimming pool.
  • CIt is the time taken by the algae to cover that area on July 5.
  • DIt is the number of days needed for the algae to cover the bottom of the swimming pool.
  • EIt is the area in square millimeters covered by the algae on July 5.

What does 2๏‘‰๏Žข mean?

  • AThe area covered by the algae triples every two days.
  • BThe area covered by the algae doubles every three days.
  • CThe area covered by the algae doubles every day.
  • DThe area covered by the algae doubles every third of a day.
  • EThe area covered by the algae triples every day.

Q22:

A mathematical model predicts that the population of a city, ๐‘ฅ million, will be given by the formula ๐‘ฅ=2(1.22)๏Š, where ๐‘› is the number of years from now. What does the model predict the population will be in 2 yearsโ€™ time?

  • A1.4884 million
  • B2.9768 million
  • C3.4884 million
  • D4.88 million

Q23:

The population of a city is growing according to the equation ๐‘ฅ=9(1.79)๏Š, where ๐‘ฅ is the population in millions, and ๐‘› is the number of years since 2015. What was the population of the city in 2015?

Q24:

The population of Malawi, in millions, between 1960 and 2016 can be modeled by the function ๐‘ƒ(๐‘ก)=3.62๏€น1.029๏…๏. By how much has the average rate of growth changed from the period 1960 to 1965 to the period 2011 to 2016? Give your answer to the nearest thousand.

Q25:

In 1970, the world population was 3.682 billion and showed a growth rate of 2.08% per year. Assuming a constant growth rate, what would have been the estimate for the size of the population in 2017? Give your answer accurate to three decimal places.

  • A4.059 billion
  • B9.689 billion
  • C9.787 billion
  • D9.890 billion
  • E8.119 billion

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