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AP Calculus AB: 11.2.3 Radioactive Decay

Mathematics10 CardsCreated 3 months ago

This content explains how radioactive decay follows exponential decay laws, using the same mathematical model as exponential growth but with a negative rate. It includes examples involving half-life calculations, solving for unknown quantities, and forming differential equations to describe decay processes.

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Radioactive Decay

  • Radioactive substances decay over time. The time required to decay by half is called the half-life.

  • The same equations that govern exponential growth apply to radioactive decay.

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Key Terms

Term
Definition

Radioactive Decay

  • Radioactive substances decay over time. The time required to decay by half is called the half-life.

  • The same equations that ...

note

  • While exponential growth involves increases that are
    proportional to the original amount, radioactive decay
    involves decreases that a...

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now. What was the amount of the material 30 years ago?

10^9/980^2

A radioactive material is known to decay at a yearly rate of 0.3 times the amount at each moment. Which of the following differential equations describes the rate of radioactive decay?

dP/dt=−0.3P

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There are 2000 grams of the material right now. Its half-life is known to be 1600 years. What is the amount right after t years?

2000 ⋅ 2^−t / 1600

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now. What will be the amount of the material right after 20 years?

980^3/10^6

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AP Calculus AB: 11.2.3 Radioactive Decay
TermDefinition

Radioactive Decay

  • Radioactive substances decay over time. The time required to decay by half is called the half-life.

  • The same equations that govern exponential growth apply to radioactive decay.

note

  • While exponential growth involves increases that are
    proportional to the original amount, radioactive decay
    involves decreases that are proportional to the original
    amount. Therefore, they use the same formula.

  • Consider modeling the radioactive decay of radium-226. Start with 100 mg as an initial quantity.

  • The half-life of radium-226 is 1590 years, so after that many years you will have 50 mg left.

  • Plug in these known values so you can solve for the
    constant k.

  • Now you can plug the value of the constant into the formula.

  • This formula can be simplified since it involves both
    logarithmic and exponential expressions.

  • Factor out the log expression.

  • The properties of logarithms allow you to make t/1590 the exponent of the logarithm.

  • Now e ln(1/2) can be reduced to 1/2.

  • Finally, the fraction can be inverted by introducing a negative sign in the exponent.

  • The negative sign in the exponent indicates that the quantity will shrink. This agrees with the principle of radioactive decay.

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now. What was the amount of the material 30 years ago?

10^9/980^2

A radioactive material is known to decay at a yearly rate of 0.3 times the amount at each moment. Which of the following differential equations describes the rate of radioactive decay?

dP/dt=−0.3P

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There are 2000 grams of the material right now. Its half-life is known to be 1600 years. What is the amount right after t years?

2000 ⋅ 2^−t / 1600

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now. What will be the amount of the material right after 20 years?

980^3/10^6

A radioactive material is known to decay at a yearly rate of 0.3 times the amount at each moment. There are 1000 grams of the material right now. Which of the following equations describes the amount of material as a function of time?

P = 1000 e ^−0.3t

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 2000 grams of the material 10 years ago. There are 1990 grams right now. What is the half-life of the material?

10 ln2/ln(2000/1990)

A radioactive material is known to decay at a yearly rate of 0.04 times the amount at each moment. There are 2000 grams of the material right now. What is the amount (in grams) after 10 years?

1.3×10^3

A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There are 1000 grams of the material right now. Its half-life is known to be 1500 years. What is the amount right after t years?

P(t)=1000e^−(ln2)t/1500