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Sagot :
Answer:
Approximate population ≈ 74.36 million.
Step-by-step explanation:
The given problem involves modeling of an exponential growth or decay function of a population.
Definition:
- Exponential growth ( relative growth ) occurs when a population grows or increases exponentially by the same factor, over the same amount of time.
- Exponential decay occurs when a population decreases continuously by a constant factor, over the same amount of time.
We can model the exponential growth of a population using the following Exponential Growth Model:
- ⇒ [tex]\displaystyle\mathsf{P(t)\:=\:P_0 (1\:+\:r)^t }[/tex]
Where:
- P( t ) = population after "t" years
- P₀ = initial population
- r = relative growth rate; positive "r" value means that the population is increasing; negative "r" value implies that the population is decreasing.
- t = time (typically in years)
Solution:
Based from the given equation, [tex]\displaystyle\mathsf{y\:=\:58.7(1.03)^t }[/tex] , we can infer that:
⇒ [tex]\displaystyle\mathsf{P(t)\:=\:P_0 (1\:+\:r)^t }[/tex] or [tex]\displaystyle\mathsf{y\:=\:P_0 (1\:+\:r)^t }[/tex]
Where:
- P( t ) or y = population after "t" years
- P₀ = initial population = 58.7
- r = relative growth rate = 0.03 or 3% = the population is increasing (exponential growth).
- t = time (typically in years) = 8 years (difference between January 2002 and January 1994).
Step 1: Substitute the given values into the Exponential Growth Model formula, and solve for P( t ) :
[tex]\displaystyle\mathsf{P(t)\:=\:P_0 (1\:+\:r)^t }[/tex]
⇒ [tex]\displaystyle\mathsf{P(8)\:=\:58.7 (1\:+\:.03)^8 }[/tex]
Step 2: Follow the order of operations, addition inside the parenthesis:
⇒ [tex]\displaystyle\mathsf{P(8)\:=\:58.7 (1.03)^8 }[/tex]
Step 3: Follow the order of operations, applying the exponent into the parenthesis (do not round off the digits inside the parenthesis):
⇒ [tex]\displaystyle\mathsf{P(8)\:=\:58.7 (1.266770081) }[/tex]
Step 4: Follow the order of operations, multiplying 58.7 (P₀) into the parenthesis:
⇒ [tex]\displaystyle\mathsf{P(8)\:\approx \:74.35940378\quad or \quad 74.36\:\:million}[/tex]
Final Answer:
Therefore, the approximate population of the country in January 2002 is 74.36 million.
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Keywords:
Exponential functions
Exponential growth
Exponential decay
Exponential growth model
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Learn more about exponential functions here:
https://brainly.com/question/25802424
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