Questions: A population numbers 12,000 organisms initially and grows by 16.1% each year. Suppose P represents population and t represents the number of years of growth. An exponential model for the population can be written in the form P = a b^t, where P = □

A population numbers 12,000 organisms initially and grows by 16.1% each year. Suppose P represents population and t represents the number of years of growth. An exponential model for the population can be written in the form P = a b^t, where
P = □

Transcript text: A population numbers 12,000 organisms initially and grows by $16.1 \%$ each year. Suppose $P$ represents population and $t$ represents the number of years of growth. An exponential model for the population can be written in the form $P=a b^{t}$, where \[ P=\square \] Question Help: Video 1 Video 2

Solution

Solution Steps

Step 1: Identify the Initial Population and Growth Rate

The problem states that the initial population is 12,000 organisms and the growth rate is $16.1\%$ per year.

Step 2: Convert the Growth Rate to a Decimal

To use the growth rate in the exponential model, we need to convert the percentage to a decimal. The growth rate of $16.1\%$ is equivalent to $0.161$ in decimal form.

Step 3: Determine the Base of the Exponential Model

The base $b$ in the exponential model $P = a b^t$ represents the growth factor. Since the population grows by $16.1\%$ each year, the growth factor is $1 + 0.161 = 1.161$.

Step 4: Write the Exponential Model

Using the initial population $a = 12,000$ and the growth factor $b = 1.161$, the exponential model for the population is:

\[ P = 12,000 \times 1.161^t \]