Question

The function y = 58. 7(1. 03)^t gives a country’s population, y (in millions), where t is the number of years since January 1994. According to this function, what was the approximate population of the country in January 2002

Answer 1

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:

• ⇒   $\displaystyle\mathsf{P(t)\:=\:P_0 (1\:+\:r)^t }$

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, $\displaystyle\mathsf{y\:=\:58.7(1.03)^t }$ , we can infer that:

⇒   $\displaystyle\mathsf{P(t)\:=\:P_0 (1\:+\:r)^t }$ or  $\displaystyle\mathsf{y\:=\:P_0 (1\:+\:r)^t }$

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 ) :

$\displaystyle\mathsf{P(t)\:=\:P_0 (1\:+\:r)^t }$

⇒   $\displaystyle\mathsf{P(8)\:=\:58.7 (1\:+\:.03)^8 }$

Step 2: Follow the order of operations, addition inside the parenthesis:

⇒   $\displaystyle\mathsf{P(8)\:=\:58.7 (1.03)^8 }$

Step 3: Follow the order of operations, applying the exponent into the parenthesis (do not round off the digits inside the parenthesis):

⇒  $\displaystyle\mathsf{P(8)\:=\:58.7 (1.266770081) }$  

Step 4: Follow the order of operations, multiplying  58.7 (P₀) into the parenthesis:

⇒  $\displaystyle\mathsf{P(8)\:\approx \:74.35940378\quad or \quad 74.36\:\:million}$

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:

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