Problem Descriptions and Solutions
- Charlie's Birdhouses Growth
- Karen's Christmas Party Guests
- Uranium-234 Decay
- Ornithine Decarboxylase Decay
Problem Description: Charlie's Birdhouses experiences an annual growth rate of 10%. In 2010, Charlie made 120 birdhouses. Determine the number of birdhouses he will make in 2015.
Solution: The exponential model for the number of birdhouses (Xj) each year is given by
X_(j+1)=1.1×X_j
And X_2010=120. Therefore, X_2015=120×〖1.1〗^5=193.26
Rounding to two decimal places, the expected number of birdhouses in 2015 is approximately 193.
| Year | Number of Birdhouses |
| 2010 | 120.00 |
| 2011 | 132.00 |
| 2012 | 145.20 |
| 2013 | 159.72 |
| 2014 | 175.69 |
| 2015 | 193.26 |
Table 1: Number of Birdhouses vs. the year
Problem Description: Karen's Christmas party started in 1990 with 12 guests, increasing by 5% every other year. Find the expected number of guests for her 2012 Christmas party.
Solution: The number of guests (Nj) follows a geometric progression
N_j=1.05×N_(j-1)
The number of guests for 2012 (2012N2012) is found to be 35.1.
| Year | No. of Guests |
|---|---|
| 1990 | 12 |
| 1991 | 12.60 |
| 1992 | 13.23 |
| 1993 | 13.89 |
| 1994 | 14.59 |
| 1995 | 15.32 |
| 1996 | 16.08 |
| 1997 | 16.89 |
| 1998 | 17.73 |
| 1999 | 18.62 |
| 2000 | 19.55 |
| 2001 | 20.52 |
| 2002 | 21.55 |
| 2003 | 22.63 |
| 2004 | 23.76 |
| 2005 | 24.95 |
| 2006 | 26.19 |
| 2007 | 27.50 |
| 2008 | 28.88 |
| 2009 | 30.32 |
| 2010 | 31.84 |
| 2011 | 33.43 |
| 2012 | 35.10 |
Table 2: Number of guests vs. the year for party guests
Problem Description: Uranium-238, when mined for nuclear reactors, produces Uranium-234 with a half-life of 24 days. Calculate the remaining amount on December 31, 2011, if there were initially 24.5g on January 1, 2011.
Solution: Using the radioactive decay formula, the remaining amount (N) is found to be 0.00064g on December 31, 2011.
N = N_0× (1/2)^(t/T)
N = N0 * (1/2)^(t/T)= 24.5 * (1/2)^15.21= 0.00064 g
Problem Description: The enzyme ornithine decarboxylase in the human body has a half-life of 11 minutes. If 131mg is present, determine the amount remaining after 1 hour.
Solution: Applying the radioactive decay formula, approximately 2.98mg of ornithine decarboxylase will remain after 1 hour.
N = N_0× (1/2)^(t/T)
N = N0 * (1/2)^(t/T)= 131 * (1/2)^(60/11)≈ 2.98 mg
Therefore, after 1 hour, approximately 2.98 mg of ornithine decarboxylase will be left from the initial 131 mg.