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].
