Question 3 (1 point) \begin{tabular}{|l|l|l}\hline Listen \end{tabular} An endangered species has a population of 5000 . Scientists estimate that the stock is decreasing at \( 3 \% \) per year. At this rate, approximately how many years will it be before \( 60 \% \) of the species remains?

To find out how many years it will take for the population of the endangered species to decrease to 60% of its current population, we can use the formula for exponential decay: \[ P(t) = P_0 \times (1 - r)^t \] where: - \( P(t) \) is the population at time \( t \), - \( P_0 \) is the initial population, - \( r \) is the rate of decrease (expressed as a decimal), - \( t \) is the time in years. Given: - Initial population \( P_0 = 5000 \), - Rate of decrease \( r = 3\% = 0.03 \), - Final population \( P(t) = 60\% = 0.60 \times 5000 = 3000 \). We need to find the value of \( t \) when \( P(t) = 3000 \). Substitute the values into the formula and solve for \( t \): \[ 3000 = 5000 \times (1 - 0.03)^t \] \[ 0.6 = (0.97)^t \] To find \( t \), we need to take the logarithm of both sides: \[ \log(0.6) = \log((0.97)^t) \] \[ \log(0.6) = t \times \log(0.97) \] \[ t = \frac{\log(0.6)}{\log(0.97)} \] Now, we can calculate the value of \( t \) using the given values. Calculate the value by following steps: - step0: Calculate: \(\frac{\log_{10}{\left(0.6\right)}}{\log_{10}{\left(0.97\right)}}\) - step1: Convert the expressions: \(\frac{\log_{10}{\left(\frac{3}{5}\right)}}{\log_{10}{\left(0.97\right)}}\) - step2: Convert the expressions: \(\frac{\log_{10}{\left(\frac{3}{5}\right)}}{\log_{10}{\left(\frac{97}{100}\right)}}\) - step3: Use the logarithm base change rule: \(\log_{\frac{97}{100}}{\left(\frac{3}{5}\right)}\) - step4: Rewrite the expression: \(\log_{\frac{97}{100}}{\left(3\right)}-\log_{\frac{97}{100}}{\left(5\right)}\) It will take approximately 16.77 years for the population of the endangered species to decrease to 60% of its current population.