Physics final exam problem: Am-241 smoke detector replacing

Physics 205B Final Exam, spring semester 2012
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Comprehensive Problem 29.79

The following claim was made on an online insurance company blog[*]:

"Smoke alarms have a shelf life, the older they are, the less effective they become. A date is printed, or stamped on the back of smoke alarms and 10 years after that date, experts recommend buying a new one... According to the Kentucky Injury Prevention and Research Center, by the time a smoke alarm is 10 years old, it has a '30% chance of alarm failure.'"

Assume that this refers to an ionization smoke detector[**] which uses radioactive (241,95)Am (half-life of 433 years). Quantitatively discuss whether or not 10 years would be a plausible time for "30% chance of alarm failure" in terms of the reduction of (241,95)Am activity. Show your work and explain your reasoning using properties of radioactive decay.

[*] theagentron.com/blog.php?id=89 3.
[**] wki.pe/Americium_smoke_detector#Ionization.

Solution and grading rubric:

• p:
  Correct. Sets up exponential or half-life exponent decay formula to solve for activity R at t = 10 years, using R₀ = 1, and finds that activity decreases to 0.984·R₀, which is a 1.6% (or approximately 2%) decrease, concluding that 10 years is an insufficient amount of time for a "30% chance of alarm failure" due solely to reduction in radioactivity.
  
• r:
  Nearly correct, but includes minor math errors.
  
• t:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors.
  
• v:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner.
  
• x:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y:
  Irrelevant discussion/effectively blank.
  
• z:
  Blank.
  

Grading distribution:
Section 30882
Exam code: finalsM0k
p: 16 students
r: 3 students
t: 0 students
v: 1 student
x: 2 students
y: 2 students
z: 0 students

A sample "p" response (from student 1009), using the half-life decay equation, and also solving for when the activity (here using "N" instead of "R") would actually decrease by 30%:
A sample "y" response (from student 2427):