Worksheet: Exponential Growth and Decay

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:

The population, ๐‘, of a city in year ๐‘ก is given by the formula ๐‘=10(2.6)๏Šช๏๏Šฑ๏Šจ๏Šฆ๏Šฆ๏Šง. Determine the year in which the population of the city was 8 million.

Q3:

Jacob invests in a savings account. After ten years, the value of his investment had doubled. What was the annual rate of interest? Give your answer to one decimal place.

Q4:

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๏. Emma says this means that the growth rate is 8.4% per hour. Her friend Jennifer, however, says that the hourly growth rate is 8.76%. Who is right?

  • AJennifer
  • BEmma

Q5:

The value of a car falls by 36% over 2 years. By considering a suitable exponential function, find the equivalent annual rate of depreciation that would produce the same fall in value over two years.

  • A 2 5 %
  • B 2 0 %
  • C 6 0 %
  • D 4 0 %
  • E 1 2 %

Q6:

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?

  • A 5 0 %
  • B 1 0 %
  • C 2 5 %
  • D 7 5 %
  • E 1 5 %

Q7:

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.

  • A ๐‘‰ ( ๐‘ก ) = ๐‘ƒ ๏€ป 1 โˆ’ ๐‘Ÿ 1 0 0 ๏‡ ๏
  • B ๐‘‰ ( ๐‘ก ) = ๏€ป ๐‘ƒ โˆ’ ๐‘Ÿ 1 0 0 ๏‡ ๏
  • C ๐‘‰ ( ๐‘ก ) = ๐‘ƒ ๏€ป ๐‘Ÿ 1 0 0 ๏‡ ๏
  • D ๐‘‰ ( ๐‘ก ) = ๐‘ƒ ๏€ป 1 + ๐‘Ÿ 1 0 0 ๏‡ ๏
  • E ๐‘‰ ( ๐‘ก ) = ๏€ป 1 โˆ’ ๐‘Ÿ 1 0 0 ๏‡ ๏

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.

Q8:

A man deposited 8,694 LE in a bank account with an interest rate of 6% per year. Determine how much money was in the account 10 years later, given that the interest was compounded annually. Give your answer correct to two decimal places.

Q9:

A man deposited 3,049 LE in a bank account with an interest rate of 10% per year. Determine how much money was in the account 7 years later, given that the interest was compounded every 4 months. Give your answer correct to two decimal places.

Q10:

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 correct to two decimal places.

Q11:

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 a daily growth rate of 1.3% per day, which is higher than ๐ตโ€™s growth rate of 0.4% per day. On which day will ๐ดโ€™s population surpass that of ๐ต?

  • ADay 1
  • BDay 32
  • CDay 31
  • DDay 407
  • EDay 2

What are the populations of ๐ด and ๐ต on the day found in the previous question? To use the model, you must round to the nearest integer.

  • A ๐ด = 1 , 2 4 9 , ๐ต = 1 , 2 3 8
  • B ๐ด = 1 , 2 4 0 , ๐ต = 1 , 2 4 3
  • C ๐ด = 4 , 1 7 3 , ๐ต = 6 , 2 6 0
  • D ๐ด = 1 , 8 4 0 , ๐ต = 1 , 3 9 5
  • E ๐ด = 1 , 8 6 4 , ๐ต = 1 , 8 6 3

Q12:

In 1859, Thomas Austin imported 24 wild rabbits from England and released them into the wild of southern Australia to be hunted for sport.

Consider that female rabbits only breed during the year following their birth and that the rabbit population is equally shared between male and female rabbits. With a birth rate of roughly 20 rabbits per female per year, by what factor would the whole rabbit population increase per year?

What was the rabbit population after 5 years?

By what percentage did the rabbit population increase per month?

  • A 2 1 %
  • B15%
  • C3%
  • D50%
  • E18%

How many months did it take the rabbit population to reach past one thousand?

Q13:

In the US, the proportion of waste that was recycled has roughly tripled between 1985 and 2005. Using an exponential model for this proportion, find in which year the proportion of waste had roughly doubled with respect to the value in 1985.

Q14:

A population of seagulls grows from 75 to 102 in 6 months. Find the continuous growth rate. Give your answer as a percentage to one significant figure.

  • A 0 . 0 5 % per 6 months
  • B 5 % per 6 months
  • C 5 % per month
  • D 0 . 0 5 % per month

Q15:

There is a gap of 3 mm between the floor and one of the legs of a table. How many times would a sheet of paper of thickness 0.08 mm need to be folded to fill that gap?

Q16:

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

Q17:

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

Q18:

The value of a car depreciates by 13% every year. If a car cost $75,000 when bought new, how old would it be when its value has dropped to $40,000? Give your answer in years to two decimal places.

Q19:

The population of bacteria found in raw milk cheeses was found to increase by a factor of 10 after 10 hours at a temperature of 33โˆ˜C.

If the population started at 50 bacteria, how long would it take it to reach 300 bacteria, assuming exponential growth? Give the answer in hours and minutes.

  • A47 minutes
  • B17 hours and 47 minutes
  • C60 hours
  • D7 hours and 47 minutes
  • E24 hours and 47 minutes

How long would it take for the number of bacteria to double?

  • A6 hours
  • B8 hours
  • C2 hours
  • D3 hours
  • E10 hours

Q20:

A population that obeys an exponential growth rate of 2.43% per year grows by a fixed percentage every two years. What is this percentage?

  • A 6 . 4 7 %
  • B 4 . 9 2 %
  • C 3 . 1 4 %
  • D 5 . 9 8 %
  • E 4 . 8 6 %

In what period will this population grow by 50% approximated to one decimal place?

Q21:

A certain growth of cancerous cells increased from 400 initially to 440 one month later. How many months will it take for the initial number of cancer cells to double? If necessary, round your answer to one decimal place.

  • A1.1 months
  • B7.3 months
  • C10 months
  • D6 months

Q22:

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.

  • A 2 . 4 3 %
  • B 1 2 . 7 2 %
  • C 1 . 1 2 %
  • D 2 . 7 2 %
  • E 7 . 8 6 %

Q23:

The cougar population in Oregon was estimated to be 5,000 animals in 2006, having grown from about 3,150 at the start of 1996. Assuming exponential growth, in what year would we expect a population of 7,500?

Q24:

A school had a wasp infestation. Around 10โ€Žโ€‰โ€Ž000 wasps were found on the school premises. An extermination company came at 9 am and treated the school with a chemical, which caused the wasp population to halve every 10 hours. The head teacher decided that the pupils would return to school on the first day when the total number of wasps on the premises at 9 am was less than 50. How many days was the school shut for?

Q25:

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.

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