Exploring Exponential Models in Real-World Scenarios

In this Data analysis assignment, we delve into the dynamics of exponential growth and decay through practical scenarios. From the projection of birdhouse production to estimating Christmas party guests and understanding radioactive decay, these examples illustrate the application of mathematical models in various contexts. Let's explore the solutions step by step and witness the power of exponential models in capturing real-world phenomena.

Charlie's Birdhouses Growth

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

Karen's Christmas Party Guests

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

Uranium-234 Decay

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

Ornithine Decarboxylase Decay

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.