10 calculus word problems (decay rates)

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NUC-490 Written Assignment 4 (Module 5) Page 8 of 10

Student Name:

Module 5 – Assignment Exercises

The following 10 exercises build upon your previous education and training, and the Summary Examples you studied from this Module. These exercises will challenge you to put all those elements together.

Exercise 1 - Given that Beryllium-11 () has a half-life = 13.81 sec,

(a) what percentage of a sample of will remain after 25 seconds?

(b) What percentage will have decayed?

1. N(t)/N0 = (0.5)^(t/13.81)

N(25)/N0 = 0.5^(25/13.81) = 0.285

a) Hence 0.285*100% = 28.5 % remains after 25 s.

b) 100 - 28.5 = 71.5 % would have decayed.

Exercise 2 – A certain radioactive material decays at a rate proportional to the amount present. Initially there is 50 milligrams of the material present. After two hours, the amount of radioactive material has lost 10% of its original mass.

a) Calculate the value of the decay constant, λ.

b) Find the half-life of the material.

c) 2. a) dx/dt = -λx

(1/x)dx = -λdt

ln x = -λt + ln c

ln (x/c) = -λt

x = c exp(-λt)

At t = 0, x = 50 mg. This gives c = 50 mg.

At t = 2 hours, x/c = 0.9. This gives e^(-2λ) = 0.9 or λ = 0.052

So, x = 50 exp(-0.052*t)

b) 25 = 50 exp(-0.052*t)

t = 13.157 hours

Semi-Log plot of radioactive decay 10

1 Number of radioactive atoms N X 10 20 0 6 5 4 3 2 1 2 3 4 5 6 8 7 9

Time (hours)

Exercise 3

Use the Semi-Log Plot of Radioactive Decay above to determine the half-life and the decay constant of the sample.

(a) Choose two points and calculate the decay constant and half-life from the exponential decay equation.

(b) Check your answer by estimating the half-life directly from the graph.

Exercise 4

The attenuated intensity I of a neutron beam normally incident on a certain shielding material is proportional to the intensity of the beam as it passes (the distance x) through the shield, i.e.

Where m is the constant of proportionality.

If a 25 cm thick shield of this material attenuates the beam by a factor of 1,000, what thickness of the material is required to attenuate the beam by a factor of 5,000? Assume the incident beam has an intensity of Io when it enters the shield.

Exercise 5

Given a radioactive particle that encounters a shielding barrier, it is known to decelerate and come to rest within the shielding in 6 microseconds (6 μs). In this example, presume the distance (d in microns (μm)) the particle will travel in the time period after hitting the shielding is given by the following equation.

Given the bounds t0 = 0 μs and tmax = 6 μs

(a) What is the maximum distance the particle will travel into the shielding?

(b) What is the velocity of the particle at t = 4 μs?

Exercise 6

Given the velocity function of a particle to be

(a) Find the position function, s(t), that describes where the particle will be at time t in microseconds (μs).

(b) Given that s(0) = 0 microns (μm), find s(6 μs).

Exercise 7

Using the log-log graph below, determine the maximum range of:

1. 1.0 MeV beta particles passing through air Ans:

1. 3.0 MeV beta particles passing through water Ans:

1. 0.8 MeV beta particles passing through plastic Ans:

1. 2.0 MeV beta particles passing through lead Ans:

Air 1000 Penetration Ability of Beta Radiation

Water Plastic or Lucite Concrete Glass Aluminum Iron Copper Lead 5 0.2 0.1 0.3 2 3 0.5 10 1.0 0.001 0.01 100 10 1 0.1 Maximum Range of Beta Particles (inches)

Energy (MeV)

Exercise 8 - Using the graph below:

a. Determine the percentage of Cobalt-60 gamma ray

transmitted through a 50 cm thick concrete shield. Ans:

b. Estimate the thickness of concrete required to attenuate

Radium-226 photons to 0.02% of the source strength. Ans:

100% Transmission of Gamma Radiation through Concrete

2 Legend 198 Au 192 Ir 137 Cs 60 Co 226 Ra 5 4 1 3 0.001% 0.0001% 0.01% 0.1% 1% 10% 0 50 cm ( 19.7 ”) 10 0 cm (3 9. 4”) 15 0 cm (5 9 ”)

Exercise 9 - The activity of a certain radioactive nuclide is plotted on a semilog graph as a function of time. Use this graphic data to determine:

a. The half-life of this radionuclide Ans:

b. How long it will take for the activity to decrease from

100,000 disintegrations per minute (dpm) to 1000 dpm? Ans:

c. Activity vs. TimeThe activity (dpm) after 88 hours Ans:

1 10 6

60 40 0 100 20 80 Activity (dpm) 1 10 4 1 10 3 1 10 5

Time (Hours)

Exercise 10 - A pipe 10 cm in diameter contains steam at 200. It is covered with an insulating material that is 5 cm thick. This insulating material has a constant of thermal conductivity K = 0.00060 cal/cm2 sec. The outside surface of the system is 35. Find the heat loss per hour from a 2 meter length of the pipe.

x 2 – x 1 5 cm x

5 cm

(1)(1)

AxByc

=+--