Question

a population numbers 11,000 organisms initially and grows by 17.1% each year.

suppose \\(p\\) represents population, and \\(t\\) the number of years of growth. an exponential model for the population can be written in the form \\(p = a \cdot b^t\\) where

\\(p = \\)

Explanation:

⚡ Using what you learned: exponential growth and decay_model growth

Step 1: Identify the initial value

The initial population is given as \( 11,000 \) organisms. In the exponential model \( P = a \cdot b^t \), the parameter \( a \) represents the initial value when \( t = 0 \).

$$ a = 11,000 $$

Step 2: Determine the growth factor

The population grows by \( 17.1\% \) each year. Convert the percentage growth rate \( r \) to a decimal:

$$ r = 17.1\% = 0.171 $$

The growth factor \( b \) is calculated as:

$$ b = 1 + r = 1 + 0.171 = 1.171 $$

Step 3: Write the exponential model

Substitute \( a \) and \( b \) into the general formula \( P = a \cdot b^t \):

$$ P = 11,000(1.171)^t $$

Answer:

$$ P = 11,000(1.171)^t $$