Final answer:
The function N=1000-300(3^{-t}) describes a decreasing population from a starting point of 1000. To find when the population of caribou is at least 800, one must solve the inequality 1000-300(3^{-t}) ≥ 800 for the variable t, which represents time in months.
Step-by-step explanation:
Two functions, P = 3 and N = 1000 - 300(3-t), are related through transformations that include a vertical shift and a reflection across the horizontal axis, combined with an exponential decay factor. To determine when the number of caribou will reach at least 800, we need to solve for t in the equation N = 1000 - 300(3-t) ≥ 800. This leads to the inequality 200 ≥ 300(3-t), which upon further manipulations, including taking the natural logarithm, will allow us to find the value for t.
First, we simplify the inequality to 3-t ≤ 2/3. Taking the natural logarithm of both sides gives us -t ln(3) ≤ ln(2) - ln(3), which simplifies to t ≥ ln(2/3)/-ln(3). Calculating this value gives us the time in months it will take for the caribou population to reach at least 800 individuals.
Reorganizing the inequality:
- Subtract 1000 from both sides to obtain -300(3^{-t}) ≥ -200.
- Divide both sides by -300, remembering to flip the inequality sign: 3^{-t} ≤ ⅓.
- Take the natural logarithm of both sides: t ≥ \frac{\ln(\frac{1}{3})}{\ln(3)} (as the logarithm of a reciprocal is the negative of the logarithm).
- Solve for t to obtain the answer in months.
By following this process, we can algebraically determine the time it takes for the population to reach at least 800 caribou.