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Worksheet: Exponential Decay Model

Q1:

A scientist begins with 100 mg of a radioactive substance that decays exponentially. After 35 hours, 50 mg of the substance remains. How many milligrams will remain after 54 hours? If necessary, round your answer to 2 decimal places.

Q2:

A doctor injected a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there were 4.75 milligrams of dye remaining in the patient’s system. Which of the following is an appropriate model for this situation?

  • A 𝑓 ( 𝑑 ) = 4 . 7 5 1 + 1 3 𝑒 βˆ’ 0 . 8 3 9 2 5 𝑑
  • B 𝑓 ( 𝑑 ) = 1 3 𝑒 0 . 9 1 9 5 𝑑
  • C 𝑓 ( 𝑑 ) = 1 3 ( 0 . 0 8 0 5 ) 𝑑
  • D 𝑓 ( 𝑑 ) = 1 3 𝑒 ( βˆ’ 0 . 0 8 3 9 𝑑 )

Q3:

At the start of an experiment, a scientist has a sample which contains 250 milligrams of a radioactive isotope. The radioactive isotope decays exponentially, so that after 250 minutes there are only 32.0 milligrams of the isotope left.

Write the mass of isotope in milligrams, 𝑀 , as a function of the time in minutes, 𝑑 , since the experiment started. Give your answer in the form 𝑀 ( 𝑑 ) = 𝐴 𝑒 𝑏 𝑑 , rounding 𝐴 and 𝑏 to three significant figures.

  • A 𝑀 ( 𝑑 ) = 2 5 0 𝑒 βˆ’ 0 . 0 0 5 6 7 𝑑
  • B 𝑀 ( 𝑑 ) = 2 5 0 𝑒 βˆ’ 0 . 0 0 7 3 4 𝑑
  • C 𝑀 ( 𝑑 ) = 3 2 𝑒 βˆ’ 0 . 0 0 9 1 4 𝑑
  • D 𝑀 ( 𝑑 ) = 2 5 0 𝑒 βˆ’ 0 . 0 0 8 2 3 𝑑
  • E 𝑀 ( 𝑑 ) = 3 2 𝑒 βˆ’ 0 . 0 0 7 3 4 𝑑

Find the half-life of the isotope, giving your answer to the nearest minute.

Q4:

The rate at which a radioactive substance decays is proportional to the remaining number of atoms. The differential equation which can be used to describe this process follows where 𝑁 represents the number of atoms remaining after 𝑑 seconds. The proportionality constant πœ† is considered the decay constant for this process. If 𝑁 0 represents the number of remaining atoms at 𝑑 = 0 seconds, find the general solution.

  • A 𝑁 = 𝑁 𝑒 0 πœ† 𝑑
  • B 𝑁 = 𝑁 𝑒 0 βˆ’ 𝑑 πœ†
  • C 𝑁 = 𝑁 𝑒 0 𝑑 πœ†
  • D 𝑁 = 𝑁 𝑒 0 βˆ’ πœ† 𝑑

Q5:

A quantity decays in time 𝑑 years according to the formula 𝑃 ( 𝑑 ) = 𝐴 𝑒 l n 𝑏 2 3 𝑑 . If half the amount is left after 46 years, what is the value of 𝑏 ?

  • A √ 2
  • B 1 2
  • C 2 √ 2
  • D 1 √ 2 ο€Ώ √ 2 2  o r
  • E2

Q6:

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.

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

Q7:

An isotope decays at a percentage rate of 1 . 2 % per year. Which of the following is the first year when there is less than half of the isotope remaining?

  • A59 years
  • B57 years
  • C56 years
  • D58 years
  • E55 years

Q8:

The concentration of aspirin in human blood 𝑑 hours after the intake of a normal dose 𝑐 0 can be modeled by the function 𝑐 = 𝑐 ο€Ό 1 2  0 𝑑 3 .

What is the half-life of aspirin, that is the time it takes for half of the initial dose to be eliminated?

  • A 6 hours
  • B 2 hours
  • C 1 3 of an hour
  • D 3 hours
  • E 1 2 an hour

Q9:

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, 𝐢 , in their system 𝑑 hours later can be found using the equation 𝐢 = 3 0 ο€Ό 1 2  ( ) 𝑑 5 . 7 .

Write the equation in the form 𝐢 = 𝐴 ( 𝑏 ) 𝑑 , giving values to 3 decimal places if necessary.

  • A 𝐢 = 3 0 ( 0 . 0 1 9 ) 𝑑
  • B 𝐢 = 3 0 ( 2 . 3 8 7 ) 𝑑
  • C 𝐢 = 1 5 ( 5 . 7 ) 𝑑
  • D 𝐢 = 3 0 ( 0 . 8 8 5 ) 𝑑
  • E 𝐢 = 1 5 ( 0 . 1 7 5 ) 𝑑

Q10:

While William was making popcorn, he noticed that the corn kernels did not all pop at the same time, and that half of the kernels popped after 2 minutes from the time the first kernel popped.

What is the formula for the number of unpopped kernels ( 𝑁 ) after 𝑑 minutes, with 𝑁 0 being the initial number of corn kernels?

  • A 𝑁 = 𝑁 ο€Ό 1 2  0 βˆ’ 𝑑 / 2
  • B 𝑁 = 𝑁 ο€Ό 1 2  0 2 𝑑
  • C 𝑁 = 𝑁 ο€Ό 1 2  0 𝑑
  • D 𝑁 = 𝑁 ο€Ό 1 2  0 𝑑 / 2
  • E 𝑁 = 𝑁 ο€Ό 1 3  0 𝑑 / 2

How long did it take William to prepare the popcorn if he used 100 kernels?

  • A about 14 minutes
  • B about 5 minutes
  • C about 18 minutes
  • D about 7 minutes
  • E about 10 minutes

Q11:

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 isotope’s quantity is then reduced by half every 5 7 3 0 years. Let the amount of the isotope after 𝑑 years be 𝐴 ( 𝑑 ) .

Write an equation relating 𝐴 ( 𝑑 ) to 𝐴 ( 𝑑 + 5 7 3 0 ) .

  • A 𝐴 ( 𝑑 ) = ο€Ό 1 2  𝐴 ( 𝑑 + 5 7 3 0 )
  • B 𝐴 ( 𝑑 + 5 7 3 0 ) = ο€Ό 1 4  𝐴 ( 𝑑 )
  • C 𝐴 ( 𝑑 ) = ο€Ό 1 4  𝐴 ( 𝑑 + 5 7 3 0 )
  • D 𝐴 ( 𝑑 + 5 7 3 0 ) = ο€Ό 1 2  𝐴 ( 𝑑 )
  • E 𝐴 ( 𝑑 + 5 7 3 0 ) = ο€Ό 1 3  𝐴 ( 𝑑 )

Write expressions for 𝐴 ( 𝑑 + 1 1 4 6 0 ) and 𝐴 ( 𝑑 + 1 7 1 9 0 ) .

  • A 𝐴 ( 𝑑 + 1 1 4 6 0 ) = ο€Ό 1 4  𝐴 ( 𝑑 ) , 𝐴 ( 𝑑 + 1 7 1 9 0 ) = ο€Ό 1 8  𝐴 ( 𝑑 )
  • B 𝐴 ( 𝑑 ) = ο€Ό 1 4  𝐴 ( 𝑑 ) , 𝐴 ( 𝑑 ) = ο€Ό 1 8  𝐴 ( 𝑑 + 1 7 1 9 0 )
  • C 𝐴 ( 𝑑 + 1 1 4 6 0 ) = ο€Ό 1 3  𝐴 ( 𝑑 ) , 𝐴 ( 𝑑 + 1 7 1 9 0 ) = ο€Ό 1 9  𝐴 ( 𝑑 )
  • D 𝐴 ( 𝑑 + 1 1 4 6 0 ) = ο€Ό 1 2  𝐴 ( 𝑑 ) , 𝐴 ( 𝑑 + 1 7 1 9 0 ) = ο€Ό 1 7  𝐴 ( 𝑑 )
  • E 𝐴 ( 𝑑 ) = ο€Ό 1 2  𝐴 ( 𝑑 ) , 𝐴 ( 𝑑 ) = ο€Ό 1 9  𝐴 ( 𝑑 + 1 7 1 9 0 )

Write a formula relating 𝐴 ( 𝑑 + 5 7 3 0 𝑛 ) to 𝐴 ( 𝑑 ) for a positive integer 𝑛 .

  • A 𝐴 ( 𝑑 + 5 7 3 0 𝑛 ) = ο€Ό 1 4  𝐴 ( 𝑑 ) 𝑛
  • B 𝐴 ( 𝑑 + 5 7 3 0 𝑛 ) = ο€Ό 1 3  𝐴 ( 𝑑 ) 𝑛
  • C 𝐴 ( 𝑑 + 5 7 3 0 𝑛 ) = ο€Ό 1 2  𝐴 ( 𝑑 ) 𝑛
  • D 𝐴 ( 𝑑 ) = ο€» 𝑛 2  𝐴 ( 𝑑 + 5 7 3 0 𝑛 )
  • E 𝐴 ( 𝑑 ) = ο€» 𝑛 2  𝐴 ( 2 𝑑 + 5 7 3 0 𝑛 )

Assume that, every 5 7 3 0 2 years, the carbon-14 isotope is reduced by the same ratio π‘Ÿ . By writing 5 7 3 0 as 5 7 3 0 2 + 5 7 3 0 2 , what is π‘Ÿ ?

  • A π‘Ÿ = ο„ž 1 2
  • B π‘Ÿ = ο„ž 1 3
  • C π‘Ÿ = 1 2
  • D π‘Ÿ = 1 3
  • E π‘Ÿ = 1 4