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Katya is a ranger at a nature reserve in Siberia, Russia, where she studies the changes in the reserve's bear population over time.The relationship between the elapsed time , in years, since the beginning of the study and the bear population , on the reserve is modeled by the following function.In how many years will the reserve's bear population be ? Round your answer, if necessary, to the nearest hundredth.years
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Q. Katya is a ranger at a nature reserve in Siberia, Russia, where she studies the changes in the reserve's bear population over time.The relationship between the elapsed time , in years, since the beginning of the study and the bear population , on the reserve is modeled by the following function.In how many years will the reserve's bear population be ? Round your answer, if necessary, to the nearest hundredth.years
- Set up equation: Set up the equation to solve for . We are given the function and we want to find the time when the bear population is . So, we set up the equation .
- Divide and isolate: Divide both sides of the equation by to isolate the exponential term.This simplifies to .
- Take logarithm: Take the logarithm of both sides to solve for . We can use the logarithm base to make calculations easier since the base of the exponential is .
- Apply property of logarithms: Apply the property of logarithms that . Using this property, we get:
- Solve for t: Solve for t.To find t, we divide both sides by -0.05＂).\(\newline\$t = \frac{\log_2(0.4)}{-0.05}\)
- Calculate value of \(\newline\)\(t\): Calculate the value of \(\newline\)\(t\) using a calculator.\(\newline\)\(\newline\)\(t \approx \log_2(0.4) / -0.05\)\(\newline\)\(\newline\)\(\newline\)\(t \approx -2 / -0.05\) (since \(\newline\)\(\log_2(0.4)\) is approximately \(\newline\)\(-2\))\(\newline\)\(\newline\)\(t \approx 40\)
