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Worksheet: Applications of Exponential Decay

Q1:

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 𝑟 rounds of the game.

  • A 𝑁 = 4 2 5 6 𝑛
  • B 𝑁 = 4 2 1 6
  • C 𝑁 = 6 4 1 4 2
  • D 𝑁 = 4 2 5 6
  • E 𝑁 = 6 1 4 2

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

Q2:

Two cars are bought at the same time. One of them costs $ 2 7 0 0 0 and depreciates at 7 % each year. If the second car costs $ 3 9 0 0 0 , at what rate does it depreciate if they have the same value after 5 years? Give your answer to one decimal place.

Q3:

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 𝑃 = 𝑎 𝑏 𝑡 , where 𝑃 is the population of black rhinos 𝑡 years after 1970. Round your values of 𝑎 and 𝑏 to 3 decimal places if necessary.

  • A 𝑃 = 6 5 0 0 0 ( 𝑡 ) 0 . 8 6 5
  • B 𝑃 = 6 5 0 0 0 ( 1 . 8 6 5 ) 𝑡
  • C 𝑃 = 6 5 0 0 0 ( 𝑡 ) 1 . 8 6 5
  • D 𝑃 = 6 5 0 0 0 ( 0 . 8 6 5 ) 𝑡
  • E 𝑃 = 5 6 2 2 5 𝑡

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?

Q4:

The production of a gold mine is 4 945 kg per year. A mathematical model predicts that yearly production will go down by 1 3 % every year. What does the model predict the production will be in 7 years time? Give your answer correct to two decimal places.

Q5:

A radioactive isotope has a half-life of 250 years. To the nearest 10 years, how long will it take for the substance to be at one-third of its original strength?

Q6:

A fast-food chain wishes to reduce the amount of fat in its burgers. If it reduces the amount of fat at a rate of 5 % every year, how long will it take to reduce the fat content by one quarter? Give your answer in years to two decimal places if necessary.

Q7:

The number of marine organisms in a pool, 𝑦 , after 𝑛 weeks is given by the formula 𝑦 = 5 6 3 1 1 2 𝑛 1 . How many marine organisms will there be in the pool after 4 weeks? Give your answer to the nearest whole number.

Q8:

Every day, following treatment with a weed killer, the area of clover in a garden is reduced to one-third of the previous day’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 𝐶 , the area of clover in the garden after 𝑡 days.

  • A 𝐶 = 4 0 ( 𝑡 ) 1 3
  • B 𝐶 = 1 3 ( 4 0 ) 𝑡
  • C 𝐶 = 1 3 ( 𝑡 ) 4 0
  • D 𝐶 = 4 0 1 3 𝑡
  • E 𝐶 = 4 0 3 𝑡

Q9:

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 1 0 % 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.

Q10:

A jewelry store has a stock of 100 necklaces. Each week it sells 5 % of its remaining stock. Write an equation for the number of necklaces 𝐽 ( 𝑤 ) that are left after 𝑤 weeks.

  • A 𝐽 ( 𝑤 ) = 1 0 0 ( 0 . 5 ) 𝑤
  • B 𝐽 ( 𝑤 ) = 1 0 0 ( 0 . 0 5 ) 𝑤
  • C 𝐽 ( 𝑤 ) = 1 0 0 ( 𝑤 ) 0 . 0 5
  • D 𝐽 ( 𝑤 ) = 1 0 0 ( 1 0 . 0 5 ) 𝑤

Q11:

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 5 0 0 bacteria, how many would there be after 10 hours? Give your answer to the nearest 100 bacteria.

Q12:

The value of a car depreciates by 1 8 % every year. Write an equation that can be used to find the value in dollars, 𝑉 , of a $ 7 5 0 0 0 car 𝑡 years after it was purchased.

  • A 𝑉 = 7 5 0 0 0 ( 1 . 1 8 ) 2 𝑡
  • B 𝑉 = 7 5 0 0 0 ( 0 . 1 8 ) 𝑡
  • C 𝑉 = 7 5 0 0 0 ( 8 2 ) 2 𝑡
  • D 𝑉 = 7 5 0 0 0 ( 0 . 8 2 ) 𝑡
  • E 𝑉 = 7 5 0 0 0 ( 1 8 ) 𝑡

Q13:

The half-life of caffeine in blood is between 3 and 7 hours, depending on the individual’s metabolism. The half-life is the time it takes for half of the initial dose to be eliminated.

Assuming an exponential model, find an expression for the amount of caffeine 𝑄 in the blood plasma after 𝑡 hours given an intake 𝑄 0 , considering the half-life of caffeine is 5 hours.

  • A 𝑄 = 𝑄 0 . 5 0 𝑡
  • B 𝑄 = 𝑄 0 . 5 0 𝑡 5
  • C 𝑄 = 𝑄 0 . 5 0 5 𝑡
  • D 𝑄 = 𝑄 0 . 5 0 𝑡 5
  • E 𝑄 = 𝑄 0 . 5 0 𝑡 1 5

What is the percentage of the hourly metabolic rate of caffeine in a human body?

  • A 1 3 %
  • B 8 7 %
  • C 0 . 5 %
  • D 1 5 %
  • E 5 %

Using what you found in the second part, what equation is equivalent to the one found in the first part?

  • A 𝑄 = 𝑄 1 3 0 𝑡
  • B 𝑄 = 𝑄 0 . 1 3 0 𝑡
  • C 𝑄 = 𝑄 0 . 8 7 0 𝑡
  • D 𝑄 = 𝑄 0 . 8 7 0 5 𝑡
  • E 𝑄 = 𝑄 0 . 1 3 0 5 𝑡

In 15 hours and 15 minutes, how long after intake are 3 4 of the caffeine dose in a drink metabolized?

  • A 2 hours and 4 minutes
  • B 5 hours and 16 minutes
  • C 2 hours
  • D 29 minutes
  • E 4 minutes

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 2 0 % . They plan to achieve their target in 4 years.

Write an equation they could use to find 𝑟 , the annual sugar reduction rate required to achieve their target.

  • A 𝑟 1 0 0 = 0 . 8
  • B ( 1 𝑟 ) = 0 . 8
  • C ( 1 𝑟 ) = 0 . 2
  • D 1 0 0 𝑟 1 0 0 = 0 . 8
  • E 1 𝑟 1 0 0 = 0 . 2

Q15:

A health food café decides to make its food even healthier by reducing the amount of salt in the food they serve. Their target is to reduce their salt use by 1 0 % .

They use 𝑆 g of salt annually.

Write an expression for their annual salt use in grams if they achieve their target.

  • A 1 . 1 𝑆
  • B 0 . 1 𝑆
  • C 9 0 𝑆
  • D 0 . 9 𝑆
  • E 1 0 𝑆

So that their customers do not notice, they decide to make the change gradually over 3 years.

What is the annual rate of salt reduction that will allow them to achieve their target? Give your answer to 3 significant figures if necessary.

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

Q16:

A charity noticed that donations are decreasing at a rate of 15% every 6 months. This month, they received $ 3 0 0 0 in donations. Assuming donations continue to decrease in this way, write an equation that can be used to find 𝐷 , the donations they expect to receive in 𝑡 months’ time.

  • A 𝐷 = 3 0 0 0 ( 1 . 1 5 ) 𝑡 6
  • B 𝐷 = 3 0 0 0 ( 0 . 8 5 ) 6 𝑡
  • C 𝐷 = 3 0 0 0 ( 1 . 1 5 ) 6 𝑡
  • D 𝐷 = 3 0 0 0 ( 0 . 8 5 ) 𝑡 6
  • E 𝐷 = 3 0 0 0 ( 0 . 1 5 ) 6 𝑡

Q17:

A food manufacturer aims to reduce the sugar content of its cereal by one-fifth every year. At the moment, 100 g of cereal contains 34 g of sugar.

Write an equation that can be used to calculate 𝑆 , the quantity of sugar contained in 100 g of cereal, 𝑡 years from now.

  • A 𝑆 = 3 4 1 5 𝑡
  • B 𝑆 = 3 4 ( 𝑡 ) 4 5
  • C 𝑆 = 3 4 6 5 𝑡
  • D 𝑆 = 3 4 4 5 𝑡
  • E 𝑆 = 3 4 ( 𝑡 ) 1 5

Use your equation to calculate the amount of sugar they aim to have in 100 g of cereal 3 years from now. Give your answer in grams to one decimal place.

Q18:

The population of a rare species of butterflies is in decline and can be modeled by the function 𝑃 ( 𝑡 ) = 𝐾 ( 0 . 7 ) 𝑡 , where 𝑡 is the number of years after the first measurement.

What does 𝐾 represent?

  • Athe final value of the population
  • Bthe number of years after the first measurement
  • Cthe rate of population decline
  • Dthe population when it was first measured
  • Ethe number of times the population is measured each year

Write an expression for the population 3 years after it was first measured.

  • A 0 . 3 4 3 𝐾
  • B 0 . 2 1 8 7 𝐾
  • C 3 . 7 𝐾
  • D 2 . 1 𝐾
  • E 0 . 2 3 3 𝐾

Calculate the percentage fall in population over the 3 years.

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

Q19:

Carbon-11 has a half-life of 20.334 minutes. At the beginning of an experiment, there were 80 grams of carbon-11 in a sample. Write an equation that can be used to calculate 𝐶 , the number of grams of carbon-11 that remain 𝑡 minutes after the experiment begins.

  • A 𝐶 = 8 0 ( 2 0 . 3 3 4 ) 1 2 𝑡
  • B 𝐶 = 8 0 1 2 2 0 . 3 3 4 𝑡
  • C 𝐶 = 8 0 1 2 𝑡 2 0 . 3 3 4
  • D 𝐶 = 8 0 1 2 𝑡 2 0 . 3 3 4

Q20:

In 2012, Jacob and Anthony bought cars. The car that Jacob bought cost 2 7 0 0 0 USD. Anthony’s car cost him 3 9 0 0 0 USD. The table shows the values of these cars in 2012 and 2013.

Jacob Anthony
2012 27 000 39 000
2013 25 380 30 225

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

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

  • A23 857 and 23 424
  • B21 245 and 22 637
  • C26 351 and 25 457
  • D22 628 and 24 148
  • E27 186 and 29 473

Q21:

A car that costs 𝐴 dollars and depreciates at 6 % per year has a value of 0 . 9 4 𝐴 at the end of that year.

What, in terms of 𝐴 , is its value after 2 years? Give your answer to three significant figures.

  • A 0 . 9 4 𝐴
  • B 0 . 6 6 5 𝐴
  • C 0 . 8 8 𝐴
  • D 0 . 8 8 4 𝐴
  • E 0 . 6 8 3 𝐴

By what percentage does the value depreciate every two years? Round your answer to one decimal place.

How many years will it take for the value to become less than 7 0 % of the original?