Question

In a particular region, the population of barn owls is modeled by the function P(t)=300(2)
t
, where t represents the number of decades since the year 2000.

Without using a calculator, what would be the best estimate for the population of barn owls in the year 2012?
A. 589 barn owls B. 600 barn owls C. 1,109 barn owls D. 675 barn owls

Answer 1

To estimate the population of barn owls in the year 2012, we first need to determine the value of \( t \) for the year 2012. Since \( t \) represents the number of decades since the year 2000, we can calculate \( t \) as follows:

\[
\text{Year } 2012 - \text{Year } 2000 = 12 \text{ years}
\]

To convert years into decades, we divide by 10:

\[
t = \frac{12}{10} = 1.2
\]

Now, we can substitute \( t = 1.2 \) into the population function \( P(t) = 300(2)^t \):

\[
P(1.2) = 300(2)^{1.2}
\]

We need to estimate \( 2^{1.2} \). Recognizing that \( 2^{1.2} \) is between \( 2^1 = 2 \) and \( 2^2 = 4 \), we can estimate its value. Taking into account that \( 1.2 \) is closer to \( 1 \) than to \( 2 \), we can estimate that:

\[
2^{1.2} \approx 2.3
\]

Now, substituting this back into the function:

\[
P(1.2) \approx 300 \times 2.3
\]

Calculating this gives:

\[
P(1.2) \approx 690
\]

Among the options provided, the closest estimate to 690 is:

**D. 675 barn owls.**