11.2.3 Radioactive Decay

Question 1

Radioactive Decay

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

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?

Answer 3

10^9/980^2

Question 4

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?

Answer 4

dP/dt=−0.3P

Question 5

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?

Answer 5

2000 ⋅ 2^−t / 1600

Question 6

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?

Answer 6

980^3/10^6

Question 7

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?

Answer 7

P = 1000 e ^−0.3t

Question 8

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?

Answer 8

10 ln2/ln(2000/1990)

Question 9

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?

Answer 9

1.3×10^3

Question 10

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?

Answer 10

P(t)=1000e^−(ln2)t/1500