```text
Given the function:

[tex]N_{\text {year }}(t)=612 \cdot\left(\frac{2}{3}\right)^t[/tex]

Complete the following sentence about the monthly rate of change in the tiger population. Round your answer to two decimal places.

Every month, the number of tigers decays by a factor of [tex]\square[/tex].
```



Answer :

To find the monthly rate of change in the tiger population given the equation [tex]\( N_{\text{year}}(t) = 612 \cdot \left( \frac{2}{3} \right)^t \)[/tex], we start by identifying the yearly decay factor from the equation, which is [tex]\( \frac{2}{3} \)[/tex].

Given the yearly decay factor [tex]\( \frac{2}{3} \)[/tex], we want to find the equivalent monthly decay factor. Since there are 12 months in a year, we need to take the 12th root of the yearly decay factor to achieve this.

[tex]\[ \text{Monthly decay factor} = \left( \frac{2}{3} \right)^{\frac{1}{12}} \][/tex]

Once we take the 12th root of [tex]\( \frac{2}{3} \)[/tex], we find that the monthly decay factor is approximately 0.97 when rounded to two decimal places.

Thus, every month, the number of tigers decays by a factor of [tex]\( 0.97 \)[/tex].