Math Problem Statement
The population of a country dropped from 52.5 million in 1995 to 44.9 million in 2008. Assume that P(t), the population, in millions, t years after 1995, is decreasing according to the exponential decay model. After how many years will the population of the country be 2 million, according to this model?
Solution
To solve this problem, let's start by using the exponential decay model formula:
where:
- is the population at time (in years after 1995),
- is the initial population (at ),
- is the decay constant,
- is the time in years after 1995.
Step 1: Determine the Initial Population ()
From the problem, we know:
- In 1995 (when ), the population was million.
Step 2: Use Information to Find
We are given that in 2008 (which is 13 years after 1995, so ), the population was million. We can plug this into the formula:
Solving for :
-
Divide both sides by 52.5:
-
Simplify:
-
Take the natural logarithm (ln) of both sides:
-
Solve for :
Let's calculate .
Step 3: Find the Time When the Population is 2 Million
We now want to find when million. Using the same model equation:
-
Divide both sides by 52.5:
-
Simplify:
-
Take the natural logarithm of both sides:
-
Solve for :
Let's calculate .
Conclusion
The population of the country will reach 2 million approximately 319 years after 1995, or around the year 2314.
Would you like a step-by-step explanation of any part of this process, or do you have any questions?
Here are 5 related questions to consider:
- How would the model change if the population were increasing instead of decreasing?
- What factors in real life might cause the actual population decline to differ from this model's predictions?
- How can we modify this model to account for a constant rate of population decline rather than an exponential one?
- What is the half-life of the population decline based on the decay constant ?
- How would the model change if the decay constant were not constant over time?
Tip: When dealing with exponential models, always check if the assumptions (like constant rate ) make sense in the real-world context.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Exponential Decay
Logarithms
Algebra
Formulas
P(t) = P_0 e^{-kt}
k = -ln(P(t)/P_0) / t
t = -ln(P(t)/P_0) / k
Theorems
Exponential Decay Model
Suitable Grade Level
Grades 10-12
Related Recommendation
Exponential Decay Population Model: Calculating Population Drop from 1995 to 2020
Exponential Decay Model for Population Decline from 49 Million in 2000 to 45 Million in 2018
Exponential Decay Model: Country Population Decline from 1995 to 2007
Exponential Decay of Population: Model and Predictions
Exponential Decay Model for Population: Predicting Decrease Over 10 Years